One combo that can easily tackle 6 or 7 epidemic cards is Scientist/Researcher. Just stay in Atlanta, swap cards and discover cures. I think it could do 8 comfortably or almost-comfortably as well.
The deck is 48 cities plus 5 events plus 6-8 epidemics, or 59-61 cards total, of which 8 are in the opening hands. So it's an epidemic every 51/6 or 52/7 or 53/8 cards, which is 8.5, 7.42 or 6.625; in the worst case, by the second epidemic you'll have drawn 13.25 cards, let's say 10 cities and 1 event, plus your initial 18, for a total of 18 cities. Most likely you have at least two cures by then and are well on your way to a third. Only move to 3-cube cities for treatment. Note that bottom-draws are guaranteed to not have any cubes already so long as you haven't had an outbreak yet.
In the easy (6-epidemic) case you'll have 15-16 cities and 1-2 events by the second epidemic, plus the initial 8 (or 7-1), for at least 22 city cards. An even spread is ~5 of each color, so you'll probably have at least three cures. Then emergency cleanup is almost trivial.
Another combo that works well is Medic plus Operations Expert. There you do the opposite: focus almost all your energy on removing cubes and only incidentally swap cards. Building up a network of research stations will make transportation easy; the OE should build one in each of his first five turns, and probably spend cards flying to the most valuable cities (those with 3 cubes, those with two that are far away from other research centers, or perhaps in places where you need to swap cards). It'll be a much more defensive game, but I found it fairly easy. Note that with more epidemic cards, the disease is concentrated on fewer cities due to the more frequent reshuffling (at least early; you also get to higher infection rates more quickly, which pulls in the opposite direction).
I wrote a fairly extensive analysis of Pandemic. Please read the other answers to that question first; they cover more basic stuff.