Is it possible that low-resource Turing Machines can always "usually" agree with high-resource Turing Machines
Okay, I realized there's an uninteresting answer to this question via a "slow" diagonalization argument.  Let ${M_k}$ be an enumeration of the machines that run in $TIME(n^2)$.  Define $\bar{M}$ to be the machine that does the opposite of $M_k$ on all inputs of length $k$.  Then $\bar{M}$ is in $TIME(n^{2 + \epsilon})$, but there is no machine in $TIME(n^2)$ that is a $0.001$ approximation of $\bar{M}$, because there is some input length on which the two machines disagree everywhere.

I don't feel like this really captures the spirit of the question, though, and I'm going to try to think about how to re-formalize it.  Maybe I want to say something about circuits?  Like: is it possible that for all circuits on $n$ input wires, $SIZE(c)$ has a $.99$ approximation of $SIZE(c+1)$?