I'm currently running a D&D 4E game with anywhere between 14-18 players depending on the time of year. While I think we do a fairly good job of keeping the story moving, keeping combat from dominating our time and actually having a decent storyline, I'm really curious if there have been any systems specifically designed for large numbers of players or systems designed for smaller groups that provide advice for GMs of larger groups.
The simpler the system, the easier it handles larger numbers of players.
Warhammer FRP 1E, Moldvay/Cook or Mentzer D&D (or their pseudoclones and retroclones), Barbarians of Lemuria... all are simple, straightforward, and pretty fast playing, so larger groups don't slow things down.
The only ones I've seen specifically designed for 10+ players have also been designed as LARP rules. On the other hand, LARP rules work just fine for table top in most cases. WWG's Mind's Eye Theater, WEG's Star Wars Live Action Roleplaying, BTRC's Epiphany, and the L5R LARP all work just fine for tabletop as well as LARP.
Systems I've run for 10+ players include Mentzer and Allston D&D editions, Warhammer FRP 1E, Prime Directive 1E (not so good, combat is slow), L5R 2E, and AD&D2E. I've played in a CP2013 game where I was player 9...
Split the Group
For a very large D&D group, consider splitting the group into two, three, even four smaller groups that play on different nights. The two groups exist in the same world, maybe even the same locale. They hear about each other's exploits and maybe even interact. You can set up a web forum or email list for them to share stories "at the tavern."
You can also split the group and play on the same night, if you have two or more DMs. Break into separate tables.
Although I wouldn't really call any D&D version as designed for large numbers of people, I have seen a lot about older editions (ante-2e) working pretty well with larger groups, reportedly because combat is so much faster. I know that the old discussion about have a party caller came out of the tendency for larger groups.
Have you considered converting your game to some kind of live-action roleplaying (LARP) game?
LARP play reduces the reliance on GMing and increases the reliance on self-adjudicated player interaction. Thus, it's not uncommon to see LARPs that entertain dozens of players at a time. Certainly, LARP is a different kind of experience than tabletop roleplaying, but it definitely can handle a large number of players.
While dungeon-oriented, combat-oriented games like D&D don't convert well (even boffer LARPs tend to aim towards "wilderness" encounters so you can play outside; few people have access to a real-life "dungeon"), politically-oriented games convert quite well.
- White Wolf published LARP rules (Mind's Eye Theater) for World of Darkness.
Advice for playing D&D with way too many players
My other answers avoid this basic part of your question. This answer goes right to the core, assuming you're still gonna play D&D 4E and you're still gonna have one DM and one giant group of players at one table. Here's what I'd do in this situation:
Avoid combat. Seriously, a combat round takes a few minutes per player. 15 players = 30-45 minutes per round. That means 30-45 minutes between each player doing anything. Expect people to get bored and stop paying attention. A six-round combat takes 3-4 hours, which is most of your game session. Don't do it.
If you must have combat, make it meaningful. The only thing worse than waiting 30 minutes between rolling dice is not caring about it when you do. Save combats for when it's really important to the characters.
If you must have combat, make it faster. Let players roll in advance; the DM doesn't need to see their roll. Have them write their actions on an index card and hand it in at the top of each turn. Yeah, that means they their action might be invalid by the time their turn comes around. Fog of war, baby. They should have rolled better for initiative. Actions can be vague though, and the DM can interpret them liberally: "Attack the goblin" might really mean "run up to the goblin and hit him." Or compromise and have players turn in their actions before it's their turn. If you don't get a card by the time it's their turn, they "delay" by default. You snooze, you loose.
Have another player help with combat. They can track initiative order, gather action cards, track effects, and so on.
Don't let players hem and haw over actions. If you don't use a written-card system (above), when it is a player's turn, he gets 10 seconds to declare his action. If he misses the deadline, he delays one spot in the initiative order. Go to the next player. Allow each player to write a "default action" on his character sheet. If he can't decide what to do, he can choose "do nothing," "delay," or "default." A good default in D&D 4E is a basic at-will power.
End combats when they're obviously over. If the combat is obviously over, just end it. Monsters run away or surrender, or that last hit just kills the monster. If it's not clear that the PCs will win but it looks favorable and the combat is getting boring, offer the PCs an out: they can all spend 1-2 healing surges to win and end it.
Spend a lot of time doing non-combat stuff. Explore, adventure, travel, and interact. However, even role-playing and problem-solving encounters are going to be chaotic with 15 people at the table. Make sure everyone stays involved.
Split the party. I don't mean make them play with a different DM or play on a different night. I mean let both groups sit at the table with y'all, maybe even at allied purposes, but give them separate attention. Don't go more than 5-10 minutes without switching between groups.
Although there is no indication that it was designed for that purpose, Cyberpunk 2020 from R.Talsorian Games, with its system (Interlock) and combat rules (Friday night Firefight) scales really well to large numbers of players. Combat rarely lasts more than three turns (it's deadly), and quick & dirty decisions fit well with the spirit of the setting.
You know, old school D&D was often played with large groups like this. I mean White Book OD&D or Basic D&D.
You'll want to take all the rules pretty seriously. The Caller rules really are designed to deal with the large party size. That is, the players kibitz amongst themselves but do not really talk to the DM. The Caller represents the group to the DM. The DM might address individual players for dice rolling and stuff.
Also look at LARPS. These are often designed for large groups of players. They're not the same without the primary control of the GM, but you might glean some useful ideas from them.
I've had one or two successful games of The Shab-al-Hiri Roach with ten players, but the rules don't really address how to make such games successful. Rather, the scene structure and the way players can drag other players in to play NPCs happen to be useful techniques for entertaining players who aren't in the spotlight.
Silhouette-based games (Heavy Gear, Gear Krieg, etc. see DP9 site) have tactical combat (i.e. not-too-abstract) and manage to be still extremely fast - I have no experience with such a large group, but I have always been impressed by how quickly combat goes in Silhouette, so I'd definitely try that if I had to referee for such a large group.