Maximum size of minimal generating set

Definition
Suppose $$G$$ is a group that has at least one defining ingredient::minimal generating set, i.e., a generating set with the property that no proper subset of it generates $$G$$. The maximum size of minimal generating set for $$G$$ is defined as the maximum of the sizes of all possible minimal generating sets for $$G$$.

Note that minimal generating sets are also called irredundant generating sets and hence the maximum size of minimal generating set can also be called the maximum size of irredundant generating set.

The maximum size of minimal generating set is defined for any finite group.