Should be a little slow to get going, then an increasingly rapid slope (you might not be able to see the slopes transition as each integrator saturates), then as it settles down, a very long settling time, to an extremely precise level (though ultimately still limited by the offset and GBW of the amps involved).
A fascinating implication of such a network is it seemingly implies that, if stages are transitioned continuously, then one would get a continuously variable (i.e., fractional) integral. In analogy to a saturating chain of ampilfiers, the sum of which exhibits a log(Vin) curve (log base being the gain per stage).
The biggest downside is, the 'plant' (last integrator / gain stage / output) needs to be incredibly fast, because each integrator takes off a factor of 3 or more from the GBW. I think. "Very long settling time" is relative to this final bandwidth, which I suppose would have to be in the several MHz range, for an audio application (20kHz+). Which doesn't really sound so bad, at least.
Also, the settling would be in terms of not just voltage, but "jerk" (derivative of acceleration, V/s^3)! Actual voltage settling may be good or poor (I'm not sure; depends how much voltage over/undershoot is required to adjust those derivatives; and the compensation*, of course), and will take all that extra time, but we're talking asymptotic reproduction of ramps and second order curves, which means fantastic AC reproduction.
*A 4th order controller plus single pole 'plant' implies we can solve for coefficients (hopefully -- if possible) that satisfy a particular filter polynomial, such as a 5th order Bessel. That would be a sweet amplifier indeed.
Tim