With a $20 limit on the budget, I would have to agree with BillyM and go with a Photon Freedom light. Having had the opportunity to play with a wide range of LED lights, I have found few that are more convenient and reliable in that price range.
Want a much more versatile light for a few bucks more? Look into
Petzl's e+Lite. While I EDC a mini-flashlight like a NovaTac EDC-120 or a Surefire L1, I never leave home without my e+Lite. Available for under $30 it gives you a remarkably bright light that has multiple hands-free options in a small package.
In the AAA or AA battery range, an ARC-AAA light would be at the top of my list, followed by a MagLite with a NiteIze 1w LED upgrade. These two lights are at opposite ends of the size spectrum, and both of those lights exceed the $20 limit. I have NiteIze modified Mini-Lights (1w & IQ switch) in a number of my medical kits and have been very happy with them.
The ARC-AAA makes a better EDC light in my mind; it is small enough to reside on a keychain but still remarkably bright. It does lack the adjustable light levels or signaling features found on so many lights today. I have had one on my keychain for 2 years without failure on a single battery!
Another good choice (well outside of your price range) but worth mentioning is Photon’s Proton. It is brighter than just about any other light its size that runs off of a single AA battery but also has a ramping brightness level so you can find just the right brightness for the task at hand. Multiple signaling modes and a red LED to preserve night vision increase its utility. I found a single drawback to this light, one that after carrying it everyday since January of this year made me take it out of EDC service and move it into a smaller kit. This light just won’t stay off in my pocket! Nothing I tried worked, so it has been moved into a fanny-pack survival kit…