I'm pretty much ok with a term limit for president, with particulars varying for governors and other executive-office holders, but for legislatures they turn the elected officials into pass-throughs and cut-outs.
There is a huge amount of things people need to know when they are drafting legislation, moving it through the process, and passing (or defeating) it. It takes years to build up the knowledge of how to do it and what the tricks are: how to slip something in, how to spot and counter something someone else has slipped in, how to move it forward, how to stall it, how to bring people on board to support it, how to undermine support, etc. Even how to do all of the above and not be obvious about it.
It takes many years to build up that knowledge base, that institutional memory. And it's not just the politicians who need to do that, it's their staff too.
Now throw in term limits, especially tight ones: Just as people develop this institutional memory, they're kicked out... or are they? Some pols and staff may go back to what they were doing before, but others can make careers of sticking around and offering their services to the new class. And to lobbyists. "Permanent Washington" (and state capital equivalents) doesn't go away -- it's actually reinforced as pols are forced to rely on them to get anything done -- it's just moved even farther away from accountability to voters.
People who want term limits think it will make politicians more responsive, but what it really does is outsource the actual work of government to persons who are not getting paid by the voters.