The smallest group requiring more than 2 generators is $$ C_2 \times C_2 \times C_2 $$ but that group is abelian. The smallest non abelian groups requiring more than 2 generators are groups with quotient $ C_2^3 $, such as $ D_8 \times C_2 $ and $ Q_8 \times C_2 $. But these extensions of $ C_2^3 $ are still solvable.
What is the smallest finite group $ G $ which is not solvable and requires more than two generators?
This question is the same as
The smallest group with 3 generators
but with "non abelian" replaced by "non solvable"
Kenta S points out that $$ C_2^3 \times A_5 $$ is order 480 and non solvable and requires at least 3 generators (since $ C_2^3 $ is a quotient). I have a feeling that we can do better and $ C_2^2 \times A_5 $ is also minimal 3 generated (EDIT: my feeling about $ 2^2A_5 $ was wrong see the answer from ahulpke or for an explicit 2 generation of $2^2A_5$ see answer from Parcly Taxel)(Also note that ahulpke found another non-solvable group of size 480 which is minimal 3 generated, namely $2^2S_5$).