large sets of commuting linearly independent matrices
Solution 1:
Let $n$ be even. Then we can achieve $1+(n/2)^2$ by the matrices of the form $$\begin{pmatrix} a \cdot \mathrm{Id} & M \\ 0 & a \cdot \mathrm{Id} \end{pmatrix}$$ where each block is $(n/2) \times (n/2)$ and $M$ is arbitrary. This is best possible, by a result of Schur. See "A simple proof of a theorem of Schur", if you have access to JSTOR.