Xjyuk

Essentially all instruments produce overtones, which are frequencies
other than the dominant frequency of the note.

-- How do harmonics work?

The use of "essentially" there got me thinking. Are there any instruments which do not produce overtones?

I've heard it claimed that human whistling comes very close to being a perfect sine wave:

The video here seems to show only one peak on the spectrograph, supporting a nearly perfectly sinusoidal waveform.

As far as I know every instrument produces overtones. Some might think that unpitched percussion don't have overtone, but they produce them as well.

However, there are some electronic instruments, such as synthesizers (sine waves) which can be played without producing any overtones, but every acoustic instrument does.

If I'm correct the ocarina might be the instrument which come as close as possible to creating 'no overtones'. In fact, they do create overtones as well, but because of their shape, the overtones are actually many octaves above the keynote scale.

A tuning fork comes close, though amplifing it by planting it on some resonating object - a wooden table, piano case, ot try your head :-) - will add some harmonics.

The sound-producing element of a Fender Rhodes electric piano is essentially a tuning fork, though other parts of the instrument are designed to 'dirty up' the pure tone.

http://www.fenderrhodes.com/org/manual/ch1.html

The tone of a flute, especially in the higher register, is close to a sine wave.

Note that we're talking about the sustain portion of a note. Both tuning fork and flute produce much more complex sounds as a note is attacked. You could mistake a tuning fork for a flute if the attack portion of a note was chopped off. I don't think you'd confuse the two if the attack was also heard though!

This principle was put to good use in 'Hybrid Synthesisers' like the Roland D50 or Yamaha SY range. A short sampled attack was followed by a synthesised sustain and release. It combined a remarkable degree of realism and controllability with economical use of sample memory.

So your answer is: although some instruments have a sustain close to a sine wave, I can't think of one outside the test bench that lacks a more complex attack.

By far, I'm no expert in this, but here's my best shot.

Timbre is the result of a specific series of overtones sounding off louder than others. We are looking for a timbre that only has the fundamental sounding off and nothing sounding above it. I suppose anything that could produce a single sine wave would be your answer. Perhaps an organ with only one tone sounding?