Where are temporary files in Chrome stored?

On Windows this is Google Chrome cache folder:

C:\Users\<user-name>\AppData\Local\Google\Chrome\User Data\Default\Cache

However:

Google Chrome cache technique is defensive!

Although you could use YouTube downloader program or manually retrieve the YouTube video direct download link to keep a copy of the video for your mom who only like to watch them on the classic TV + player set, you might need to use non-Google Chrome web browsers to retrieve the FLV files on some less popular video hosting sites, via the web browser cache folder technique.

Read this excellent post for some more info (though it might be outdated)

On Windows XP:

C:\Documents and Settings\USERNAME\Local Settings\Application Data\Google\Chrome\User Data\Default\Cache

you can also type about:cache into your address bar to see cached files, as the names in the above directory aren't very helpful (data_0,data_1,f_00000a,f_00000a2,f_00000a3, etc...). When in about:cache you can right-click files and choose Save Link As... which should use the cache as the source.

Depending on your configuration(s), Chrome may indeed store YouTube videos in the temp directory. However, they will be locked (i.e., cannot be opened or copied), as well as set so that they are automatically deleted when the lock is released (i.e., when the tab with the video is closed). You would need to first identify the correct file (this is easy if you observe the directory while the video is buffering because you can see the filesize growing). Then you can use Unlocker to save a copy of it.

Another, easier option is to simply use an extension like YousableTubeFix or YouTube Center.