Polynomial, trigonometric, exponential and fractal curves

[Loren Booda]

What other curves are there that cannot be described by the above? Are trigonometric functions actually a special case of exponentials with complex powers?

[hypermorphism]

Since the trigonometric functions can be written as combinations of sines and the sine function can be written as a complex exponential, we have exponential and logarithmic functions covering them and their inverses.
The Weierstrass curve and other functions represented by infinite series and solutions of differential forms give an infinitude of curves that cannot be described by finite means. Proprietary functions such as the Lambert W-function (http://mathworld.wolfram.com/LambertW-Function.html) also abound. As another addition, there is also a vast jungle of nowhere analytic curves that cannot be described with any analytic function.