Going for an exotic tropical beach location? Well they normally stay in sterile 5* bubble which is boring beyond belief and very unauthentic. You can go ie to Seychelles or Bali, live with locals in cheapish airbnbs, swim on same beaches or go to same restaurants they do, if you want, or even better eat with locals too. You can dive on same spots, kite surf on same spots etc. You will remember such vacation much more than they will do.
Or another typical one - skiing holidays. You can go ie to Verbier or Chamonix and ski next to kings, princes or industry moguls on same slopes they do, use same lifts, but you can ie go of piste for some extra fun. Sure afterwards you can only go to public spa but that may be better equipped than their private one. Or you can paraglide over them (on your own, not with paid instructor). And so on and on...
I know you can outmatch them only on specific aspects of those experiences, but that part is rather easy. Rich play their game of life in general very safely, so you can have way cooler things if you take some risks, and invest time into learning how to get best out of travels and adventures. Rich generally pay others to figure these out for them, and then of course such service is not well tailored to specific personality and expectations as much.