large sets of commuting linearly independent matrices

Given a set of $n \times n$ real matrices which are linearly independent and commute with one another, how large can the cardinality of this set be? By using diagonal matrices we can have such a set of size $n$ and since diagonal matrices commute with upper triangular ones, we can get $n+1$ too. Can we do better?

What about the case of matrices over finite fields?

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