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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?

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    $\begingroup$ You should also note that it is possible to confirm gravitational time dilation without entering space. $\endgroup$ – called2voyage Nov 21 '16 at 15:41
  • $\begingroup$ Not specifically for measuring time dilation due to gravity but the Deep Space Atomic Clock will have the ability to do so. $\endgroup$ – FGreg Nov 21 '16 at 16:57
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    $\begingroup$ First done in 1959 (a tour-de-force in elegant experimental design): en.wikipedia.org/wiki/Pound%E2%80%93Rebka_experiment $\endgroup$ – dmckee Nov 23 '16 at 3:33
  • $\begingroup$ In addition to the experiments listed already in comments and answers, there were mountain/valley experiments using atomic clocks carried out by Iijima in 1978 and Briatore in 1979. Also the classic 1971 Hafele-Keating experiment aboard commercial jets en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment , as well as several more precise follow-up experiments by Alley et al. using aircraft. $\endgroup$ – Ben Crowell Nov 23 '16 at 5:47
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    $\begingroup$ Time Dilation also affects the GPS system. users.sussex.ac.uk/~waa22/relativity/… $\endgroup$ – Scott Goodgame Nov 23 '16 at 15:34
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You don't need a space probe. Or an aircraft. Or even a car. NIST has measured the predicted general relativity time dilation due to a change in altitude on Earth of one foot!

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    $\begingroup$ I read an article one time of a father and son performing an amateur experiment with two atomic clocks and a drive up a mountain to demonstrate time dilation. It was a success. $\endgroup$ – called2voyage Nov 21 '16 at 17:27
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    $\begingroup$ I think this may be the one I'm thinking of. It was actually a father with his three kids and three atomic clocks! $\endgroup$ – called2voyage Nov 21 '16 at 17:50
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    $\begingroup$ «...drove back down the mountain... That cost me $50 in gas and 1 nanosecond of time.» Reminds me of how painful it was to watch Interstellar, even though he only lost 1ns, not 23 years. $\endgroup$ – ecc Nov 22 '16 at 12:26
  • $\begingroup$ Just think what a computer could have done with that one nanosecond... ;) $\endgroup$ – Tracy Cramer Oct 15 '18 at 18:08
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In addition to specific probes like the one mentioned by called2voyage, the effect is significant enough that it affects everyday operations. For example, the GPS constellation needs regular clock corrections because the satellite hardware sits much higher up the gravity well than the ground hardware.

The Wikipedia page for gravitational time dilation mentions other confirmations in equipment flying in airplanes, and even between two points only one meter distant in height under laboratory conditions.

In general, any instrument taking precise measurements over a distance must take relativistic effects into account as well, especially if it and its target are moving at significantly different speeds or are at significantly different heights. Physics SE may be able to better answer exactly when the effects become impossible to ignore.

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Yes, time dilation was experimentally confirmed by Gravity Probe A, launched by NASA on June 18, 1976.

The clock rates of two masers (one on the probe and one on Earth) were compared, and it was found that the difference matched what was predicted with an accuracy of about 70 parts per million.

To address your question of challenges in designing the mission:

To yield an accurate and inexpensive experiment, GP-A required a flight path with a large change in the gravitational potential to provide a large gravitational redshift, and it required a flight path that kept the flight Hydrogen MASER in contact with the ground Hydrogen MASER during data collection. [Stanford]

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