A good notation and a bad notation (in my opinion).

Good: $p' = (1 - \frac1p)^{-1}$. It is commonly enough used in analysis (Holder inequality) that it is good to have a shorthand, and it makes clear that the conjugate exponents are dual pairs: $(p')' = p$.

Bad: $p^* = \frac{np}{n-p}$ the Sobolev conjugate in Sobolev inequalities. It hides the dependence on the spatial dimension $n$, and overloads the $ * $ for something that does not have a duality: $(p^* )^* = \frac{(2p)^*}{2} \neq p$.