Well, we have general purpose processors. You can prove they can run any algorithm you want (i.e. are Turing complete), but also,
for practical problems (i.e. the ones encountered in engineering solutions in our planet and in our universe), they give reasonable max-min performance. Analogously I don't think 'AGI' is entirely useless -- you'd expect an AGI to have some properties like being able to solve reasonably well problems found in nature and society, maybe have a motivational framework distinguishing it as a separate entity, some knowledge about the world, etc.
edit: In terms of Turing-completeness analogues, the best candidate for AGI I think would be simply brute force capability: can this agent try all possible solutions until it solves this problem? (obviously using a heuristic to prioritize) -- that is, it'd employ a form of Universal Search[1] (aka Levin Search). Humans don't necessarily pass this test rigorously because we'd always get bored with a problem and because we have finite memory. But then CPUs are not truly Turing complete either (it's "just" a good model).
[1] http://www.scholarpedia.org/article/Universal_search