The NiteCore T0 AAA fits your requirements nicely.

The Nitecore EZ AA is also a fine flashlight but a bit above your price range (maybe $50). It's a twisty two-stage (6 and 160 lumens). Crank up the lumens for brief periods to do things, like search under your desk for that dropped pen, but you still have super long runtime at the 6 lumen setting plus the higher capacity of the AA battery over a AAA one.

I recommended it just the other day, so I'll just say that the E05 is also a very useful light that I now EDC. Just in my own experience, an E01's output is often not quite enough and I wish I had more light, such as when you need to look for something. That's where the higher output of the E05 makes it more useful to me than my prior EDC, the E01. Of course, it loses in the runtime department to the E01.