[lagadu]

Technically Turing completeness requires infinite memory for that (or an infinite tape if we're talking about the original turing machine concept), which no Turing-complete machine has. In other words, the brain is as Turing-complete as any machine that we also consider to be so. We'll always be bounded by limited memory and limited time.

[SomeoneFromCA]

We do not know if human brain is indeed Turing-complete, or even if it is a Turing machine at all. Human Mind certainly is, but if brain is or not we do not know.