Functions which can be solutions of exponential algebraic equations.

Question

Can the equation

$p(f,e^f)=0$

have a solution in $L^{\infty}([0,1])$?

Here $p(x,y)$ is a polynomial with complex coefficients, and $L^{\infty}([0,1])$ is the space of Lebesgue measurable functions from $[0,1]$ to $\mathbb{C}$ which are essentially bounded. If so what regularity properties can the solutions have?