Donald Green in Yale's political science department has posted the replication data for many of his experiments. You can find them here:
Green is also the director for the Yale Institution for Social and Policy Studies. They have a replication data archive here:
Of course, a lot of data is also available at the ICPSR and Dataverse websites, though you will have to dig through to find experimental data. You can find them at the links below:
Depending on your interests, many of the top journals (in the social sciences, at least) have started to institute policies requiring that researchers make their data available for replication purposes. I suggest reading through several academic journals, find an article that interests you, and then look up the researcher's home page. Many of them make their data readily available, and most should be happy to share it if it's not already posted.