All the ones I know of are:
1) My personal favorite is the "atomic clocks flying around the world" test, or the Hafele–Keating experiment. Wikipedia summarizes it quite well:
The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity.
2) Similarly, this also came up when the first atomic clock was put into orbit:
At the time of launch of the first NTS-2 satellite (June 1977), which
contained the first Cesium clock to be placed in orbit, there were
some who doubted that relativistic effects were real. A frequency
synthesizer was built into the satellite clock system so that after
launch, if in fact the rate of the clock in its final orbit was that
predicted by GR, then the synthesizer could be turned on bringing the
clock to the coordinate rate necessary for operation. The atomic clock
was first operated for about 20 days to measure its clock rate before
turning on the synthesizer. The frequency measured during that
interval was +442.5 parts in 1012 faster than clocks on the ground; if
left uncorrected this would have resulted in timing errors of about
38,000 nanoseconds per day. The difference between predicted and
measured values of the frequency shift was only 3.97 parts in 1012,
well within the accuracy capabilities of the orbiting clock. This then
gave about a 1% validation of the combined motional and gravitational
shifts for a clock at 4.2 earth radii.
3) Historically, the only obvious alternative to relativity (if I may oversimplify a bit) is the idea that light waves travel through some kind of medium, which was then called the aether. The Michelson-Morley experiment was an attempt to measure Earth's movement through the aether using a clever arrangement of lights and mirrors. Despite their best efforts, they never measured any such movement, which was a strong blow against the aether theory and set the stage for relativity. These lecture notes explain the full story in a thorough but approachable way.
4) People working with particular accelerators effectively re-prove this with every experiment. One of the simplest effects you have to take into account (i.e., one of the few I can claim to understand) is that firing one high-speed particle into a stationary particle takes a lot more energy than firing two slightly-less-high-speed particles towards each other.
5) The orbit of Mercury famously does not agree with Newtonian mechanics, unless one assumes there is another planet between it and the Sun. No one ever found such a planet, because it turned out general relativity was the correct explanation.