I believe I have zero Alkaline batteries left in my house and I'm relatively surprised that pretty much everything works fine. If anything, I suspect the only problem is that some devices have an inaccurate account of how dead the batteries are. But I use Eneloops on everything, even things surely not designed at all to run on them. (And I reckon you could probably make more devices work if you really wanted to; adding an additional cell or two in series would surely give you a voltage that's in range, if you can figure out a good way to do it.)

Of course not all rechargable batteries are the same; there are a few different rechargable battery chemistries in the AA form factor. I like Eneloop Pros, though; they've been very reliable for me. I've been using them for years and I've never had to throw one out yet; supposedly they last over a thousand cycles with most of their capacity.