That is, a function that is neither even nor odd, doesn't repeat at a non-changing interval, and doesn't contain a middle (or center of three-plus- dimensional) about from which the output increases/decreases toward ∞ or -∞ with respect to the input value. The range might or might not go to either infinity or both (which if it does would suggest a vertical asymptote given that domain is not restricted).
If one exists, then what parent function form or forms can fit this? If there are multiple, then which one would be considered the simplest?
An example that is also continuous and does not converge to a limit in either direction is
$$ f(x) = (\pi+\arctan(x)) \sin (x).$$
This looks a little like a sinusoidal function, but the $(\pi+\arctan(x))$ term ensures that every cycle has a different amplitude, so they never exactly repeat and you have no odd or even symmetry, even if you translate the function sideways and vertically.