I had an idea last night:
Run ALL encounters as minions.
Seriously, why does every battle need to be a tactical chess match? Fantasy film and literature are full of battles where the heroes are just hacking through a swarm of bad guys. Also, if you WANT a more video game feel to your RP, this can simulate all those action sequences, be it from Zelda, Prince of Persia, or God of War.
Taking the OPPOSITE approach, if you like a more RP oriented game, where combat is more of a plot device than anything else, then all-minion battles can speed things a long.
Now, I don't mean that these kind of games should only include the "minions" listed in the MM. I mean that one could turn all the monsters into minions by implementing a new rule:
All stats stay the same, but HP is lowered so that each creature can only absorb 1, 2, or 3 hits.
Obviously, there would be "Boss" fights where monsters are left pretty much unchanged.
Whatever the motivation, an "all-minion" campaign would have the effect of speeding up combat, which can mean A) more time RPing, B) battles with even MORE monsters, or C) more frequent battles.
Note: I'm not an advocate for turning D&D into a video game, but this could make for an interesting change of pace.