There are a lot of incredibly fast growing functions defined on the natural numbers.
Typical examples start with tetration, further hyper operators, Ackermann, and then there is monsters like the function used to declare Graham's Number, Goodstein's function, and not ending at TREE(n) or even Busy Beaver.
It seems like a natural question for anyone fascinated with large finite numbers whether any of these functions extend to the reals in a "natural" way.
Here, by natural, various things are meant. "Continuous", for instance, seems easy if you just take line segments. But there should be better properties possible - quite possibly depending on what function we talk about. For instance, differentiability - it seems likely since we can always find a faster growing real analytic function. Can we actually always find a real analytic matching function?
For things like tetration and, more generally, hyper operators, laws like we have for exp() would be something that would seem natural - but is any of this possible?
Now, even the Wikipedia article on tetration shows that we really cannot expect quite as nice properties as for exponentiation, but it does not go into detail on the possibilities of achieving some sort of reasonable laws for the extended function.
Based on the really good arguments here, perhaps it makes more sense to restrict oneself to positive reals, though maybe not?
Lastly: It seems to me as if somehow "naturally" defined functions on $\mathbb{N}$ grow fast, real functions should allow more possibilities. So, are there real functions known to grow faster than the above mentioned on the natural numbers?
Any function on $\mathbb N$ (or any other subset of $\mathbb C$ with no limit point) can be extended to an entire function. This is a consequence of Weierstrass's and Mittag-Lefler's theorems. I don't know about "natural".