What are the odds of 2 14500 “3.6V” lithium ion batteries being able to replace 4 AA “1.5V” batteries?
Assuming the two 14500s would start out at 4.2V, that would add up to ~8.4V, which is above what the AAx4 would start with (~6.4V?)
I know that if the max voltage the 4-AA-designed-circuit could handle was 7V, then 8.4V would damage it.
But if I don’t know what the circuit can handle, what are the odds? Do manufacturers often give more grace above 6.4V for 4 AA systems? Has anyone tried this?
If those AA’s are in parallel then the device is still using !.5 volts.
If they are in series the device will see 6ish volts.
2 14500’s in parallel will get you 4.2 volts.
2 14500’s in series will get you over 8 volts.
If this is a light your working on and the AA’s are in a carrier, sometimes you can remove the battery carrier and replace it with a 18650 or 26650. Just depends on the device you are working on.
As far as higher voltages hurting devices, well, sometimes the circuits can handle it and sometimes it can’t. If you are going to do it just be prepared to see the magic smoke. If no smoke, all may be well. I guess it all depends on how much you like this device, whether you should do it or not.
I’m guessing, and I’m happy to be corrected by others- you might be able to stuff a lithium cell and circuit regulator in the space, plenty of single cell boost regulator/converters on eBay etc
Shorter runtime than the AA cells though.
Why not just use rechargable NiMh cells?
Oh, good point on the rechargeables. I had two reasons in mind:
1. I find NiMh to be a pain: they seem to often go “bad” whereas my lithium batteries seem to not have any “memory” effect (so I can top them up whenever)
2. 1.2 V is the cutoff, so I can’t run 1.2V cells.
It was $6, so I may just try it. Maybe use 80% charged batteries.
Good NiMh cells won’t give you any grief. Eneloop will last you a very long time with virtually no problems. If you cheap out on your cells then you are almost guaranteed to have issues. NiCd batteries were horrible when it came to memory effect. You don’t see that with modern day quality NiMh so much.
$6? What the heck. Give it a shot. 
You didn’t mention what the mystery “circuit” is.
Without knowing, the answer is “Depends.”.