Are $10\times 10$ matrices spanned by powers of a single matrix?

There are many reasons why this is not possible. Certainly citing the Cayley-Hamilton theorem comes to mind, from which it follows that already $A^{10}$ is certainly linearly dependent on the previous powers. But there is also a very basic argument: if there were such a matrix $A$, then all $10\times10$ matrices, being polynomials in$~A$, would commute among each other. Which they don't.

Any $n \times n$ matrix $A$ satisfies $p_A(A) = 0$, where $p_A$ is the (degree n) characteristic polynomial of $A$; in particular, $A^n$ can be written as a linear combination of $I, A, A^2, \ldots, A^{n - 1}$, and thus by induction so can any higher power of $A$. So, the span over $\mathbb{F}$ of $\{I, A, A^2, \ldots\}$ has dimension at most $n$, which (for $n > 1$) is smaller than the $n^2 = \dim_{\mathbb{F}} M_n(\mathbb{F})$ asked.

For reasons explained by others no single one matrix can provide a basis for all the other matrices.

However The Jordan normal form is interesting to consider if we want to investigate the requirements for being able to express one matrix $A$ in terms of another $B$. They obviously must have at least pairwise same dimensionality of generalized eigenspaces because the block structure is preserved by any polynomial (or any other function also). Then we can try and find pairwise equivalence relations for the generalized eigenspaces. $T_iA_iT_i^{-1} = B_i$ how this would translate to requirements for a polynomial.. I don't know. But it is an interesting question!