I think I'm checking back the river.  The only worse hand that calls is QQ, maybe JJ.  That's a max of 12 combos, but in reality a lot less since QQ is something they would 4b with.  However there are a ton more Ax and suited ace rag combos that they would open flat preflop.  Pretty sick though.

You're thinking about it wrong.  Sure it's highly unlikely someone's going to flop quads like that, but that's a function of that particular flop is very unlikely (not sure if I'm doing it right, but the odds of any rank flopping trips is [4/52]*[3/51]*[2/50]=[24/132600]=1/5525, and the odds of aces specifically is [1/13]*[1/5525]=1/71,825.

But you shouldn't care how unlikely a flop is, you should only care about how his preflop range connects with that flop.  How many AdXd and AdXx vs how many TT, JJ, QQ, and KK there are.  Or to put it another way the 2nd nuts (which is technically the 3rd nuts here) does not look that appealing when there are about 20 ways to have the nuts (and 1 way to have the 2nd nuts).    Even if he is calling a river shove with TT+,66, A2dd-A5dd, A9dd+, AdJx+, you're only ahead about half the time.  I could be greatly underestimating villians ability to stack of with 88 or 77, but if he's that much of a station he's probably calling with even more aces preflop.  Even if it's chip EV it's such a marginal spot and it's for your tournament life I would just back.