The sound of inharmonicity comes from dispersion -- different frequencies travel through the string at different velocities. The lower the frequency, the less the wave notices the stiffness of the string, so to speak, and the stiffer the string the quicker a wave will travel through it. If you've ever tapped on a wire fence or played with a slinky, you'll be familiar with the "pew" sound. A pure impulse consists of all frequencies at once, but when it's traveled through the fence, bounced off a post, and come back, you hear the high frequencies first, which is why the "pew" descends in frequency.
Anyway, I've found that you can notice this on the attacks of bass notes on pianos. Only the very lowest notes of the Alexander piano seem to really have it audible, and even then it's much more slight.
I thought the cause of inharmonicity was that higher frequencies cause steeper bending of the string at the attachment points, which makes the effective length of the string shorter. Sound on Sound's article on synthesizing guitar sounds (which have the same problem) has a good diagram explaining it (see figure 12):
Thanks for sharing that article. My understanding is that this bridge effect is actually what is happening across the entire length of the string; it's just that the boundary condition makes the effects of stiffness clearer.
Stiffness causes there to basically be a radius of curvature in the string when you apply a force. The boundary condition of a guitar string is that the displacement and first derivative of displacement of the string are zero at both ends. So, this radius of curvature will be visible there. But, even when plucking a string, rather than having a sharp peak at the plectrum, it will necessarily be similarly smoothed out. (Though, through time in a frequency-dependent way.)
In the wave equation, stiffness involves a factor with a coefficient proportional to Young's modulus. Based on the stress/strain graphs I could find, Young's modulus of a guitar string increases with tension, increasing inharmonicity. Of course, the pitch of the string also increases with tension, so there's a lot going on.
(I have to admit that the zero-first-derivative boundary condition having no additional effect is coming from my intuitions about linearity of solutions to the wave equation, but maybe it still has some interesting effect. I think the overall effect of stiffness would dominate this one, however.)
I feel like the lower notes are much more distinct and separate but might just be imagining it.