A great 1x AA radio is the Sony SRF-59 FM/AM Radio Walkman (with headphones) around $16-$18. Battery is supposed to last (I would say, at least) 100-140 hours.
The other little radio that I like is a Sony SFR-M37V FM/AM/Weather/TV Radio Walkman with 25 Memory Presets (and Headphones). This is a digital radio that uses 1x AAA battery which lasts around 25-30 hours or so.
The '59 is a bare bones basic 1x AA analog radio that is very similar to an earlier version that I have used for around 10 years while mowing lawns and working. I have gone through a dozen worn out (and torn out) head phone cords and finally had the headphone socket simply wear out. Radio is sensitive and sounds good. I purchased three more for emergencies.
The 'M37V is a digital radio that is almost as sensitive as the '59. Other than the fact that it uses a single AAA battery--this radio is about perfect for a bug-out bag. It not only receives AM/FM, it also receives all eight weather stations and it can receive TV channels 2-13. All with a couple pushes of the buttons. Because it has a clock, you can't leave a battery plugged in for long-term storage. But it is pretty easy to just put the radio and a couple batteries in a plastic bag for emergencies.
Lastly, I would also suggest that you look at these Sony head phones: Sony MDR-E828LP Fontopia Earbuds with Winding Case. These cost around $10-$12 and, for me, I much prefer over standard head phones that come with the above radios. They are ear buds, very small (compared to standard over the head type phones which are bigger than the radios themselves) and it takes little power to get high volume (battery efficient). Lastly, I like them because normal headphones seem to hurt my ears (feeling of pressure) after awhile. The only drawback is that the foam earbud covers do wear out and, other than Radio Shack, I have not found another place to get replacements.
The '39 is available directly from Amazon. The 'M37V and 'E828 are available through Amazon retailers and (in my area) Target department stores.
I have crank and rechargeable/solar radios, and find that it is, most of the time, just better to have a pack of good AA and AAA batteries on hand for normal and emergency use. I go through the batteries fast enough that I always have fresh (less than 1-2 year old) batteries on hand.
-Bill