What is the best FREE PC game recording software

Really, you can get any program free (ahem, illegally), but I guess what you mean is an actual freeware program. A place to start would be either CamStudio, or Frontcam/Hypercam 2/Other low quality programs. CamStudio is the best in my opinion, but it has quite a few bugs with it. For instance, closing the program can make the ENTIRE screen flash green. The only way to solve this is to shut down the device. You will not encounter these problems with something such as Frontcam. Less features = Less problems! Another FREE alternative is to use a livestreaming service such as Twitch.TV. People love to use twitch as it is a dedicated gaming livestream hosting site thing. If you download a program called Xsplit from their site, it will allow you to livestream HOURS of footage, taking little to no computer space! A few downsides to this are: It takes a while for the video to be up on your channel. You can only record a certain amount of time and turn them in to highlights. You may lose frames depending on the quality of your internet. Livestreaming can slow your computer. Just because a game runs with no lag with nothing else running, doesn't mean it won't lag while using internet, RAM, and possibly your GPU. Another thing to note, is if you DO take up livestreaming as a whole, you may need 2 monitors, or you WILL have to run every game in window mode.
TL;DR version:Crappier freeware has less features and bugs. Higher end freeware has worse problems. Livestreaming doesn't take up space on your HDD, but it can be computer intensive and has some other downsides. If you start livestreaming, you might need more than one monitor in order to manage your stream, moderate the chat, and other things. If this can't happen, then window mode for every game is the only other option.
Hope I helped :)