Pillai's conjecture -- that the gap between (nontrivial) powers is unbounded below -- is still open (it would be a consequence of the $abc$ conjecture, were that proven). But I wonder what the right order of magnitude for it is, even though a proof seems far off.

Suppose $n=|a^x-b^y|$ for integers $a,b,x,y$ with $a,b\ge1$ and $x,y\ge2.$ What is the (conjectural) maximal order of $a^x$?

$17 = 378661^2 - 5234^3$ so we should not expect it to be too tiny. But I don't even know if the right order should be polynomial, exponential, doubly-exponential, etc. Does anyone have insight here?