I think they claim 5 years. I don't keep track of this stuff very carefully but based on how long I've had them and being on my second round of battery replacements, I'd say I got more like 3-4 years. It's not that big a deal in the grand scheme of things, especially when the batteries were cheaper. I just got sticker shock last time I went to buy a set.
Until recently at least, Energizer was the sole supplier of these batteries. You can't use anything else because they have a different voltage and discharge curve, so if you put alkaline batteries in, the Nest will almost immediately complain that they're low. I see there are some mildly sketchy other brands on Amazon now. Not sure if I'm going to take that chance.