Background

I was recently involved in a production of a play in which main character finds deceased sister's D&D binder, and takes it to a DM so she can play it. D&D monstering ensues on stage.

I've never played D&D, and I have no clue which edition was supposed to have been used. The play took place around the time Alien3 came out, so any editions released after that are not it.

Question

How feasible is it to play D&D with one DM and one player?

My question applies for all editions, including but not restricted to 5e.

I'm more interested in 5e, because if I were going to attempt it, I'd attempt in that edition. I am interested in past editions mainly just to satisfy curiosity.

Completely feasible, unless you're playing 4e

As much as people like to pretend you 'need' a Cleric or a Rogue or a Tank or whatever in the party, you really don't. This is true for every edition of D&D. Having only a single player just means you focus the campaigns you run around the character choices that player makes. If you, as the DM, say "There's a dungeon of deadly traps over there" and the player doesn't think they can deal with that they will either hire an NPC rogue or go somewhere else. Here's how things have gone with single-player parties for me, across editions:

Basic D&D:

Players die a lot. This isn't different from when they work together in a party. They die somewhat more because a single die roll is more weighty (a single lucky hit can kill everyone in the party instead of 1/4 the party). They die somewhat less because they are stealthy instead of incapable of being stealthy because someone in the party always insists upon wearing heavy armor and making a lot of noise.

AD&D 2.0:

We switched to using FATE after a bit, because the player was more interested in politics and how the city government was structured and such than dungeon crawling and I'm not going to run a political game in D&D. It was fine while it lasted, though.

D&D 3.5:

This is the system I have the most one-on-one experience with. One of us would pick a published module and run it for a character played by the other person. We'd make a character half the level that the party was supposed to be at, so that the module would be challenging. We never had any problems running said games, though the difficulty spiked significantly at higher levels of course.

D&D 4e:

This system requires the most work if you want to remake published material for a single player. It's very focused around gamified cooperative tactics and has no real support for political or investigative or stealthy games. Furthermore, the combat is wildly different with one character in the party than with a bunch; you can't have varied roles with a single character effectively, and 4e really really expects you to have a varied party composition. All of the fights are basically a cake-walk or impossible with 1 player, because they either target their strengths or their weaknesses. You don't have mechanics for things like social interaction, lock picking, magic out of combat, etc beyond the skill challenge system, which is a design feature that helps avoid those parts of the game so that you can get back to combat. Maybe it's just because I really don't like 4e as a system, but I don't think 1 player D&D is viable in that game.

D&D 5e:

Same as 3.5 except the publishers weren't completely unaware of how the game works, difficultywise. When playing with one character we use the following homebrew gestalt character rules and it works fine:

Each character gets two classes. You need the same amount of XP as normal to level up and when you do you level up both classes. No multiclassing is allowed.
You get hp from both classes, though we do a 2-player variant where you just get the better die of the two. You can still only concentrate on one spell at a time, but we're considering changing that to two except that it feels weird narratively. The interplay of abilities between your two classes makes up for the fact you only get one action per round.

Yes, it is absolutely feasible, but it is different.

Yes, I have played more than one edition of DnD with 1 player and 1 DM frequently. You mentioned that you are interested in all editions so I will address those I am familiar with. I have done this frequently using DnD 2e and lightly with 3.5e and 5e. It can work very well, though it does require some adjustments from playing with a larger party.

There are a number of ways of adjusting it to make it work better. I discuss two things that have worked well for me personally, tailoring the adventure and providing support below. Those have worked very well for me in extensive 2e play with one player and one GM, but I have less experience playing this way with 3.5e and 5e and none at all with 4e so your mileage may vary by edition.

Tailor the adventure to the character

When you have a single character for things to work well, the adventure needs to be tailored to the character's strengths. When you have a party, especially a fairly balanced one, you can generally expect that some character can come up with whatever is needed to meet any level appropriate challenge. That is much less true when playing with a single character. You will not likely have fun if the player has a single-class fighter and the adventure centers on a dungeon filled with hidden traps and locked doors. On the other hand, a single fighter dropped into a goblins den with lots of fights but few traps may have a lovely time.

Provide support

This one varies slightly by edition, but not in a fundamental way. While the terminology and techniques vary, every edition has some form of NPC support available for the players. Many tables ignore these or use them lightly when there is a traditional party, and generally I think that is a good idea. When you have a single player though it makes far more sense for them to have assistance of some kind, perhaps rotating depending on the situation. This gives the player some method to make up for the abilities that would normally be available to a party.

While I generally advise against it, a DMPC, handled as a "sidekick" rather than a full partner can work well in a 2 player game. I caution that it should be a sidekick because if the DMPC starts taking too much of the spotlight it can become boring for the player very quickly.

KorvinStarmast helpfully mentioned in a comment that there is UA material for sidekicks for 5e. I wasn't aware of that before. After looking it over, I think it could be very helpful in a 1 player game, but I have no experience with it. I use the term sidekick in the superhero sense of someone that helps the main hero while generally being less powerful, generally listening to the hero, and only rarely taking the spotlight.

Yes it's feasible; I've done solo adventures in three editions

Question

How feasible is it to play D&D with one DM and one player?

Very.

I'm more interested in 5e, because if I were going to attempt it, I'd
attempt in that edition. I am interested in past editions mainly just
to satisfy curiosity

Solo Adventures in D&D 5e

I have done this in 5e. You can find a text record of one such adventure in the Back Room chat here at RPG.SE with @Shalvenay and myself, starting roughly here. I recall that we did three sessions, but it might have been more.
My character's name is Malik Mabir, half elf, paladin. He had to discover through interactions with various NPC's (1) what the mystery / quest was, (2) what the McGuffin was, (3) who he needed to work with to return it to its rightful place, and (4) how to get out of a few sticky situations that turned into combat rather before he'd have preferred to. There was some combat, a bit of exploration and significant measure of Role Playing. There was a chase/pursuit scene as well.

• Chase/pursuit scenes are one of the bread and butter encounters to
  have for solo D&D, in my experience.

@Shalvenay also DM'd a solo adventure with Sora Tamashii in the RPG.SE back room. They did quite a few sessions. Reading through those two different adventures (Sora's was IIRC far more social/RP heavy than mine was) you can see a way to do a solo adventure. It's not the only way, but I found the adventure that I did with @Shalvenay to be a lot of fun.

FWIW: other input on solo 5e is here

Running solo adventures in AD&D 1e

I ran and played solo adventures in AD&D 1e with a friend in college circa 1978, on the weekends when we were stuck there. It happened with some frequency, we were always short of funds. We'd trade off exploring one another's worlds: his was a homebrew AD&D 1e / Chivalry and Sorcery hybrid, but the game play was AD&D 1e at its core. Mine was mostly Old D&D (three books + Greyhawk + Eldritch Wizardry) because it's what I had built to date. In those editions, when 1 GP = 1 XP, it was very beneficial to try and complete an adventure / quest / treasure hunt with less combat, not more. The payoff of treasure and XP (and thus levels) was only rewarded if one survived the quest and thus got "there and back again" to enjoy the rewards.

Fun fact: it was even feasible to play by one's self in a random dungeon by using the AD&D 1e Dungeon Masters Guide's "Random Dungeon Generation For Solo Play" (page 173). The two of us tried that a couple of times with two players versus the random dungeon. shrug Not awesome, but doable. I did it once solo. The PC died rather early in the adventure so I abandoned that pursuit. (Don't look at me like that: we didn't have MTV!)

Running solo adventures in D&D Original (three books in a box)

I did a few solo adventures as DM when I was trying to design beginner adventures in the late 1970's, playing with my brother as the adventurer. As it worked out, the one we ran for the longest that I later submitted (as a solo adventure) to TSR in the early 1980's did not make the cut. (It was not published. Such is life).

Of possibly greater interest: Rob Kuntz (you'll see his name on the OD&D Greyhawk Supplement) ran solo adventures with Gary Gygax with some frequency in the early stages of the game's history (early-to-mid 1970's). You'll see a few of his reminiscences at his web site if you are interested. His Fighting Man Robilar has a variety of tales and legends attached to him within the old school D&D gaming community. Robilar is one of the few PCs who survived the original Tomb of Horrors adventure. There is a Q&A about that here at RPG.SE.

What does a solo adventure look like?

One character using his or her wits, sometimes with an NPC or two who have been convinced via role-play (or hired) to accompany the PC on parts of the adventure. In my experience, the player is truly solo a little over half the time, and the other half of the time various NPCs are allies or hirees. The DM will need to consider that in designing the solo challenges for the player. Some of the adventure should be pure solo, but all of it need not be. (IMO, the 5e Rogue can go solo heavy, approaching pure solo, based on the class features available).

Most of the adventure scenes are not combat brawls, but include a varied combination of exploration, trap detection and avoidance, knowing when to fight and when to flee, puzzle / problem solving, and lots of role playing to find out necessary information which usually includes who to trust and who not to trust, and how to get to the McGuffin or quest object at the end.

What does a solo adventure "feel like"?

Our RPGSE friend @BlackSpike makes this very valid point: one-on-one games are very intimate. Very customised. The GM is not trying to balance a group of players. It's all about The One! True for D&D, and also true for other RPG's.

5e Sidekicks

For this edition, WoTC publised an Unearthed Arcana for Sidekicks that makes for an easy-to-insert support character for when your PC needs or wants one. I'd recommend it to any DM. I've used it to insert NPC's in my current campaign.

A rough power estimation for encounters

As a very rough rule of thumb, I have found that if I have a level 3 character versus a CR 1 monster/NPC, that's close to balanced but it really depends on the PC and the NPC. Starting out, I'd recomment CR to be 1/4th of character level until you find your sweet spot for combat encounters. This math is very much not precise.

Using the knock out rule in 5e

As a DM, one of this edition's quirks that gives you a lot of latitude in keeping the PC alive (if you so desire) is this rule:

Knocking a Creature Out

Sometimes an attacker wants to incapacitate a foe, rather than deal a killing blow. When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out. The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable. (Basic Rules p. 79, same text in PHB)

This gives the DM the chance, when the dice are cold or the PC took on a fight that was too much for them, to knock the PC out and have them captured rather than killed. (And the next adventure becomes: how does the PC escape, or talk their way out of captivity)?

In the solo adventure that @Shalvenay ran for me, he used that rule to handle my paladin being knocked to 0 HP during a formal duel.

For 5e, the essentials kit introduces a new 1-on-1 rules variant. UA (playtest content) has rules for sidekicks, that can help fill out a party.

I'm not sure if older editions had any similar rules, but for 2e I did sometimes play with only one other friend.