Tests of gravitational time dilation are also within the reach range of enthusiastic amateurs using second-hand atomic clocks:

Part 4 (starting pdf page 44) of http://web.stanford.edu/group/scpnt/pnt/PNT18/presentation_f... (slides)

Or if you prefer non-PDF:

http://leapsecond.com/great2005/

http://leapsecond.com/great2016a/