If you are looking for a real function, following is one:
$$ f(x) = e^{cos(\frac{2}{3}\pi)x} * cos(sin(\frac{2}{3}\pi)x) $$
Plot of f, f', f'', f''':
Plot of f, f', f'', f'''
You can verify it by: derivative-calculator.net
For more common problem: Function whose nth derivative is itself, following is one:
$$ f(x) = e^{cos(\frac{2\pi}{n})x} * cos(sin(\frac{2\pi}{n})x) $$
With n = 1, 2, 4:
$$ f(x)_{n=1} = e^{cos(2\pi)x} * cos(sin(2\pi)x) = e^x * cos(0) = e^x $$
$$ f(x)_{n=2} = e^{cos(\pi)x} * cos(sin(\pi)x) = e^{-x} * cos(0) = e^{-x} $$
$$ f(x)_{n=4} = e^{cos({\pi \over 2})x} * cos(sin({\pi \over 2})x) = e^0 * cos(x) = cos(x) $$
short explain:
$$
(e^{wx})' = w * e^{wx} \\
(e^{wx})^{(n)} = w^n * e^{wx} \\
$$
We known $w^n = 1$ has n roots:
$$ w = cos(\frac{2\pi*t}{n}) + isin(\frac{2\pi*t}{n}), t = 0, 1, 2...n-1 $$
let:
$$ \begin{align}
w_1 & = cos({2\pi \over n}) + isin({2\pi \over n}) \\
w_2 & = cos({2\pi \over n}) - isin({2\pi \over n}) \\
a & = cos({2\pi \over n}) \\
b & = sin({2\pi \over n}) \\
w_1 & = a + bi \\
w_2 & = a - bi \\
\end{align} $$
we got:
$$ \begin{align}
f(x) & = {{e^{w_1x} + e^{w_2x}} \over 2} \\
& = {{e^{(a+bi)x} + e^{(a-bi)x}} \over 2} \\
& = {{e^{ax} * e^{bix} + e^{ax} * e^{-bix}} \over 2} \\
& = {{e^{ax} * (e^{bix} + e^{-bix})} \over 2} \\
& = {e^{ax} * (cos(bx) + isin(bx) + cos(-bx) + isin(-bx)) \over 2} \\
& = e^{ax} * cos(bx) \\
& = e^{cos(\frac{2\pi}{n})x} * cos(sin(\frac{2\pi}{n})x)
\end{align} $$