A good question to ask!
As said by Tood Wilcox and Matthew Read, combined sounds don't just average their frequency; instead they superimpose. To get a better idea, it might help to consider water waves: you can have big shallow ocean waves and much smaller excitations on top at the same time. Clearly, the result is not just a single wave (monochromatic, to use the proper term) with the average frequency, but, well, a distinguishable combination. It works just the same way for sound waves†.
That is, except when it doesn't. Your assumption actually holds sometimes, if only under special circumstances.
When you hear two violinists playing a single note unisono, you hear in fact not two individual pitches which just happen to be the same frequency (almost the same: intonation is never completely accurate). Instead, you do hear a single note with something like the average frequency, but also a slightly different tonal character. The reason is quite a remarkable mathematical fact, namely that the additive superposition of two sine oscillations is equal to the multiplicative superposition of two other oscillations.
sin(θ) + sin(φ) = 2 · sin(θ+φ/2) · cos(θ−φ/2)
Here, θ+φ/2 is your average frequency! θ−φ/2 is the beat frequency; if θ and φ are very similar then the difference will be much smaller than the individual frequency. In fact, if one violinist plays 439 and the other 442 Hz, then the difference frequency is inaudible infrasound! The cos(θ−φ/2) factor is nevertheless not irrelevant. Multiplication by a constant like 2 means modifying the amplitude and thus also the loudness. cos(θ−φ/2) is almost constant because the frequency is so low, but nevertheless it changes in time, making the sound louder and quieter periodically. This is the phenomenon called beat.
Now, it only works quite so simple for two pure sine (or cosine, doesn't matter) waves of the same amplitude. Real instruments do not cause sine oscillations, they have more complicated waveforms, and somewhat different between any two instruments. Nevertheless, beat is audible when you tune e.g. the empty strings of two violins to match, together with other effects that make the sound generally more “smooth / silky” – I won't discuss that here.
I said special circumstances. Well, when the frequency is not almost the same, then the sine beat will be too fast to actually be recognised as loudness modulation. However, there are other special cases which lead to a similar phenomen: instead of almost the same, φ≈θ consider now twice the frequency, i.e. φ=2·θ. (Again the following will also work with almost twice, you just get an additional beat term.) Then two oscillations of φ will exactly fit into one oscillation of θ. So the result has the same periodicity as a θ-oscillation alone – by definition, this means it has the same frequency! Does this then mean that superimposing a double frequency doesn't change anything? Well, not quite. The waveform will sure be different. In fact, one useful way to interpret the different waveforms of various instruments is that they all consist of a superposition of many integer-factor sine waves, the overtones, mixed with different amplitudes. Some instruments actively exploit that: an organ creates its different sounds by mixing pipes whose frequency is a factor 2, 4, 6 or 8 apart. Each combination sounds different, but they're all perceived as one tone with a “shared frequency”.
Except when they aren't. This illusion does work for an organ passage where every note uses the same combination. If you have actual independent instrument voices, then each chord has a slightly different mixture. Our ears notice this and the brain is able to deduce back the individual frequencies. There can be other cues as well, e.g. in a strummed guitar chord the notes don't begin quite at the same time. Nevertheless, the sound of (almost) integer-related frequencies is perceived as, well, consonant, harmonious. That's how chords work!
†Actually, sound waves are simpler than water waves – these involve funny circular movements around imaginary surface points. Sound waves behave more like a queue of people (≈ air molecules) in which everybody waits to get pushed from behind, and then transmits the push to the guy standing in front (but nobody actually steps forward). You could then imagine an ongoing sequence of shoulder-taps every few seconds, interspersed by two-hand pushes every minute or so. Again, superposition of two frequencies. Of course that's also not really how sound waves behave, but you get the idea.