One of my crowning achievements as a GM is running a teen superhero game for about 10 people and having it feel like there was a max of 5 players in it. When I see it felt like 5 players I don't mean I ignored half the table. I mean that the level of depth into character's personal lives, goals, and influence was that of a much smaller game. Now the game was amazing, and I am at best only 40% of what made it awesome (with 10 players, I may even only be 20% or less) but it's a game that people still talk about, and I've had people ask about handling large groups, so I figured I'd go into it.
Without further ado, go beyond the break for some quick tips on running larger games.
Tip 1: Be Prepared
This comes across as a "no shit, sherlock" advice, but I don't mean be prepared for the game. Well, I mean, I guess that too. What I mean though is be prepared for the number of players. In general, the workload on the GM goes up multiplicatively with each additional player, not linearly (I think that makes sense...) The increase in work going from 3 PCs to 4 is greater than that of going from 2 PCs to 3. Go above 6 players and you're past the threshold most people agree is the "max" for a good game with solid looks into who the PCs are. Go even further? And yeah, you have your work cut out for you. So be prepared for that. Running the game is likely going to be more mentally fatiguing than you're used to, and that's alright.
Tip 2: Focus Your Sessions
With 7+ players you need to have a plan going into every session, and part of that plan should be "what gets the main focus of the session, what gets secondary focus, and what gets tertiary focus?" Just like a team comic book, or a show with an ensemble cast you need to have it broken down. Is the over-arcing plot the main focus or the secondary? What characters are going to have their stuff get the spotlight? Finally, what are you going to do to keep the other players involved and having fun?
Now, RPGs are great for this in a sense because players are frequently conditioned to work as a group. This means that you can do some "group stuff" for everyone, go a bit deeper with things for 1-3 of the PCs, and move the plot along for the whole game. Now who the 1-3 PCs that get focus should rotate through the group at an even rate, but you shouldn't be trying to cram major personal stuff for all 7+ PCs into one session unless you have something ridiculous like a 12 hour session planned. Even then I'd advise against it.
Tip 3: Don't Expect The Impossible From Yourself
As a GM your job isn't to provide all the entertainment, but it is to manage the entertainment and the spotlight. Don't go in expecting the impossible from yourself. With 7+ players you have a full plate, so don't expect the session to go perfectly, and don't expect to give tons of focus or session time to every player on an individual basis. That is impossible, and the only thing that lies that way is madness.
Tip 4: Delegate What You Can
Like I said, you have a lot on your plate so delegate what you can. Do you have a big combat going on? Maybe have one of the players track the initiative order for everyone. That's one less thing you have to do. Did a PC (or a couple) go off for an important task and you need a bunch of NPCs in the scene? Pass those NPCs out to the non-involved players. Give them a card with information on who they're playing, and go with it. It keeps the Players whose PC isn't on screen involved with the game and gets you a different view on an NPC that adds depth to the game. Those NPCs that are super important to the story can come with you, but having regularly assigned NPCs can also be fun for people.
Oh, also, another trick for big combats? You have 10 NPC attacks to handle? Have your players roll the dice for it. Just tell everyone to attack the person on their left, or right, and give them what they add to the roll. You get the numbers a lot faster and can essentially resolve 10 attacks in almost no time.
Tip 5: Watch Out For Special Snow Flake Characters
Every player should be trying to make an awesome character, and that's fine. However, with 7+ PCs at the table you want to watch out for what I call special snow flake characters. These are characters that go against the grain of the game, and are problematic from a conceptual level. Most large groups have at least one person with an idea like this, and while as a GM you should minimize saying no to characters you have to remember that it is up to you who you allow in your game.
So what do I count as a special snow flake character? Some examples would be the player wanting to bring a Gaijin into your L5R court game set in Otosan Uchi before the Scorpion Clan Coup. A player wanting to be an anti-hero, or a villain in a game about super heroes. A player wanting to be a tavern owner in a game about wandering the country and exploring dungeons. Effectively, the more work you as the GM have to do to not only get the PC involved in the game, but also remain a part of the game, the more likely they are to be a special snow flake and not worth letting in to the big game. Even Wolverine and Batman are teamplayers in the X-Men and Justice League books, so should your players.
Tip 6: Make Sure You Are Having Fun
Finally, do what you need to to make the game fun for you. With 7+ PCs and a GM you have at least 8 players sitting around the table. EMotions are going to catch and if you're stressed out and frustrated your players are going to read that. Make the game fun for you. That's not an excuse to wipe the party out to prove you're amazing, but do things that you find awesome and fun. Throw a PC through a wall, destroy a city block, go nuts and zany with plot elements. You're a player in this game too, and the more fun you're having with the NPCs, Encounters, and Plots, the less your PCs are going to realize that they've spent more time playing an NPC this session than their own character and the more they're just going to be having a blast.