The Busy Beaver is a classic example of a function which is not computable by a Turing Machine.
Are you really saying that a computer with too little memory can't do it, or something like that? because it seems blatantly obvious that a computer can simulate a turing machine.
If you're talking about the halting problem now, humans also don't know whether the TM they are manually simulating will halt eventually.
And you still haven't said specifically what the thing is that the humans do and computers can't.