Algorithm with no closed-form exact complexity Does there exist an algorithm for which an exact complexity provably cannot be expressed in closed-form? 

Here closed-form means a finite composition of addition, subtraction, product, division, factorial, power with any exponent, logarithm, trigonometric function, inverse trigonometric function, hyperbolic function, and inverse hyperbolic function. You may choose a subset of the above functions to allow in a closed-form expression; this makes the problem easier. However, the larger the set of allowed functions, the better, since this also answers the problem for the subsets. 

Exact complexity is a function from the input-set to a real number. You may group the input by some property, and then study exact worst-case complexity instead (or exact best-case complexity).

Any computational model will do, as well as counting any resource (e.g. number of comparisons). To close off a trivial solution, a function without a closed-form expression cannot be a primitive operation of the computational model.

If yes, is there a simple example of such an algorithm?
The runtime of an algorithm that computes the Ackermann function can't be expressed using primitive recursive functions. All commonly known named functions (well, besides the Ackermann function) are primitive recursive as far as I know.