Which registry key holds the license type in ArcGIS 10.x?

If its a 64bit machine it stores it under...
HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\ESRI

Here's ESRI's page on changing your license level without using ArcGIS administrator:

http://support.esri.com/en/knowledgebase/techarticles/detail/24633

The license level can be set using an environment variable. Of course, without admin rights you can't change environmental variables in the Control Panel, but you can do it from the command line without admin rights. Here's a little .BAT file I made for this purpose:

REM At the end of the next line, use "Viewer" for ArcView, "Editor" for ArcEditor and "Professional" for ArcInfo.
SET ESRI_SOFTWARE_CLASS=Viewer
"C:\Program Files (x86)\ArcGIS\Desktop10.0\Bin\arcmap.exe"

You would change the text at the end of the second line depending on which license level you want, and change the path in the third line depending on the version of ArcGIS you're running and the exact path to your ARCMAP.EXE file (or ARCCATALOG.EXE or whichever program you want to run). Changing the license level via this method is temporary, it will only use the license level you specify for the ArcMap session you're launching in the third line of the batch file. If you launch ArcMap in another way (via its Start Menu entry, or by double-clicking an MXD, etc.) it will launch with the license level specified in ArcGIS Administrator.