Why are all non-polynomial functions are basically exponents? There's paucity of really "original" functions in Math. Aside from power functions/ polynomials, really the only other function widely used is exponential. 
For example, $\log$ is simply inverse of $\exp$, $\sin$ and $\cos$ (and, by extension, other trig functions) can be obtained using Euler's formula, and so on. 
Why is this? Is there any philosophical reason why Mathematics did not come up with other, different basic "building blocks" of functions?
 A: I would say that the most widely used functions are simply solutions to differential equations. Consider the following differential equations:
$$y^{(n)} = C$$
$$y' = y$$
$$y'' = y$$
All of these differential equations are relatively simple and the solutions to these equations are $n$th degree polynomials, exponential functions and sum of exponents (including sine and cosine). Furthermore, many "small" changes to these differential equations ($y' = x + y$) produce solutions that are the sum or product of polynomials and exponentials. I recommend that you plug in some other differential equations that just involve derivatives of $y$ and see how often they involve exponential solutions. (See the comment about eigenvalues above).
I would say that the most common phenomena grow proportionally to their value and are best represented with exponents. If not, they may grow proportionally to their second derivative. There are many other differential equations such as
$$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0$$
which define the Bessel Function but I would conjecture that this type of relationship is less common than something such as exponential growth. If it was common then it would probably belong to the standard canon of functions. Similarly for the Airy function.
The Gamma function is widely used but it is an extension of the factorial which very simple recursive definition which could explain why it pops up so much.
I would like to note that rational functions and logistic functions should be added to the list of "non-polynomial important functions." 
A: 
Is there any philosophical reason why Mathematics did not come up with other, different  basic "building blocks" of functions?

No. The reasons are practical. There are almost no uses in practice for tetration and other  higher-order operations, so exponentiation, which is at its basis a generalization of repeated  multiplication, is the last operation of practical interest to humans. Stirling's formula, for  instance, which is helpful for approximating factorials and, by extension, combinations or  binomial coefficients, would be an exception to the rule, and there are perhaps a few others,  such as, say, Bell numbers and the like. The former is of the order $n^n=~^2n$, and the latter  of the order $e^{e^x}$.
