Mark Adler wrote in an answer to a different question that
We would expect that Voyager 1's local time is faster than Earth time by about one part in one hundred million.
I don't think that the Voyager oscillator is stable enough to measure that small of a difference.
This effect would be due to gravitational time dilation.
Given that "all" that is needed would appear to be a highly stable and accurate oscillator and counter, it seems a somewhat simple thing to test that would not add a lot to the mass or power budgets of a spacecraft.
Have we ever launched any probe that included an experiment to prove or disprove gravitational time dilation? If yes, then what were the results thus obtained?
Particularly if not, then are any such probes currently known to be in the idea or planning stages? Also, what are the major obstacles in designing and launching such a probe?