The class of recursive (total) functions is not recursively enumerable, i.e. not acceptable by a Turing machine. This result can be found in any textbook on recursive function theory (ask me off list and I'll dig up a reference).
You lost me there. I thought recursive class is a subset of recursively enumerable class. Perhaps you are referring to a special case.
More interestingly, it is possible to (uniformly) define classes of total functions that are strictly larger than the primitive recursive class - in particular, this is what type theorists spend a lot of time doing. (See, for example, Simon Thompson's book on Type Theory and Functional Programming.)
I have skimmed through the following work. http://www.math.chalmers.se/~bove/Papers/phd_thesis_abs.html She claims that it is not possible syntactically detect termination, if the recursion is not based on smaller components at each turn. But I think she doesn't supply a proof for that.
Now, a question of my own: is it possible to uniformly define a class of functions that have a non-trivial decidable halting predicate? The notion of "uniform" is up for argument, but i want to rule out examples like Andrew's that involve combining classes in trivial ways.
You lost me once again. What do you mean, could you please eloborate?