It depends on what level of gameplay you are talking about. For the casual player, this wouldn't actually change much, since in game you can usually muscle through things with STABs and the like, and a coverage move or two. It would make it easier to have HMs though, since you could add an HM into a say fifth moveslot.
Competatively, even adding a fifth move wouldn't end well. Many Pokemon can get by counters by running niche moves, but then lose coverage. Add in a fifth moveslot, and suddenly, you don't have that problem. Add a sixth one, and things get even crazier. Any more than six and things are just getting over the top. By that point, some mons don't need coverage moves, and can start running boosting moves instead.
Long story short, I could see adding a fifth moveslot working our decently. Adding a sixth one is iffy at best, and more than that should not under any circumstance be done.