There have been some excellent teams mentioned (and some that make me :facepalm1
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As for a response to the OP, I've still got to say that the best tag team ever was the Freebirds (Hell, they named a rule after them). They could all work, they could all electrify the crowd, they could all play face or heel (really depended on what part of the country they were in; in the South, they were almost always faces, everywhere else, heels). They were just an amazing tag team. The Road Warriors are a close second of all time. I give the Freebirds an edge because there were three of them and only two guys in the Road Warriors.
Recent memory (Attitude Era and since):
1 - Edge and Christian
2 - Harlem Heat
3 - Hardys
4 - Dudleys
5 - APA
Honorable Mention - BoD (would be in the Top 5 if they were a more regular tag team), New Age Outlaws, DX, Outsiders, Beer Money, MCMG, Briscoe Brothers, Haas-Benjamin/The World's Greatest Tag Team, Castagnoli-Hero/Kings of Wrestling.
wk