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Suppose that $f\left(z\right)$ is a meromorphic or entire function satisfying $P\left(z,f,f^{\prime },\mathrm{\dots },f^{\left(n\right)}\right)=0$ where $P$ is a polynomial in all its arguments. Is there a limitation on the growth of $f$, as measured by its characteristic $T\left(r,f\right)$? In general the answer to this question is not known. Theorems of Gol'dberg, Steinmetz and the author give a positive answer in certain cases. Some illustrative examples are also given.