Regular languages that can't be expressed with only 2 regex operations
With only union and concatenation, you can't describe any infinite language. The union and concatenation can only produce finitely many strings. With only union and the Kleene star, you can't describe a language such as $L = \{ab\}$, because there's no way to concatenate an expression generating only $a$ with an expression generating only $b$.  With only concatenation and the Kleene star, you can't describe a language such as $L = \{a,b\}$.