TITLE: Are there complexity classes with provably no complete problems?
QUESTION [24 upvotes]: A problem is said to be complete for a complexity class $\mathcal{C}$ if a) it is in $\mathcal{C}$ and b) every problem in $\mathcal{C}$ is log-space reducible to it.  There are natural examples of NP-complete problems (SAT), P-complete problems (circuit-value), NL-complete problems (reachability), and so on.
Papadimitrou states that "semantic" complexity classes like (I think) BPP are less likely to admit complete problems. Then again, we do not actually know whether P = BPP or not, and if so, there would be BPP-complete problems. (There exist IP-complete problems, since IP=PSPACE, and determining whether a quantified boolean formula is satisfiable is PSPACE-complete.)

Question: Are there any natural complexity classes that can be shown not to have complete problems?

I think this question has to be modified slightly, because I would imagine $Time(n)$ has no complete problems because log-space reductions can introduce a polynomial factor. 
So, in the word "natural," I include the assumption that the complexity class should be invariant under polynomial transformations (I don't know how to make this precise, but hopefully it is clear).
(Also, time or space bounds should be at least $\log n$, of course.)
Edit: A commenter has pointed me to an interesting result of Sipser that $\mathrm{BPP}^M$ for $M$ a suitable oracle does not have complete problems.  Is the same true for a (less fancy) class of the form $\bigcup_f TIME(f)$, where the union is over a class of recursive functions $f$ that are all polynomially related to each other? (Same for $\bigcup_f NTIME(f)$, and ditto for space.

REPLY [4 votes]: This is more of a comment than an answer, but the comment go too long. From this thread, there seem to be two different themes to coming up with classes without complete problems. 
Completeness is defined using two properties. L is X-complete if
(1) L is in X
(2) L is X-hard (under some suitable notion of efficient reductions)
The first theme involves classes which have hard problems, but if the hard problem were a member of the class itself, it would cause problems. The examples of POLYLOGSPACE and ELEMENTARY fall in this category. Both have hard problem, of course, but if the hard problem were a member of the class, some hierarchy theorem would be violated. (Space hierarchy and time hierarchy theorems, respectively.) Similarly one could come up with more examples of this kind.
The second theme involves classes which have no hard problems, such as ALL or P/poly. These classes don't have a complete problem for a fundamentally different reason than the previous case.
It would be interesting to see if there are other classes which fail to have complete problems for completely different reasons.