Solomon Draft is very satisfying. Gather three booster packs per player and shuffle them all together. Take turns revealing seven cards (excluding the five basic land types) and splitting them into two* piles. Then your opponent chooses a pile and you get the remaining pile.
* More with more players.
Trying to balance 1v4 or 2v3 cards with power levels as varied as Magic cards rigorously tests your evaluation skills. Choosing between two piles of very closer power level—while considering what colors and synergies you've got so far, and that your opponent has so far—is a tough and very fun challenge.
The downside is that it's very easy to have both players end with half of each color and build 3+ color decks with insufficient fixing. Also, both players have seen every card, which hurts tricks and bluffing.
Winston Draft is for smart gamblers. Gather three booster packs per player and shuffle them all together. There will be three draft piles next to the deck. On your turn, add a card to the first draft pile, look at the pile, and decide whether you want to keep the entire pile or not. If so, the turn passes. Otherwise, do the same thing with the second pile. If you put that pile back too, do the same thing with the third pile. If you put the third pile back, take the top card of the deck and the turn passes. Always look at the piles in the same order.
Winchester Draft I'd forgotten about until writing this article. You shuffle your own three packs and each turn, every player puts two cards from their stack into two draft piles in front of them and the active player chooses a draft pile to take. This will play a lot like Winston Draft but entirely public. Because you see all possible piles, you don't have to worry about the possibility of passing up a decent pile in hopes of a better pile but getting stuck with dross.
Again, it's really cool how the worst pile will get better and better thanks to the sheer quantity of cards. Again, luck will often put the best new card into that pile to negate that coolness. When I try this, I'd like to try letting each player choose which of the two piles in front of them they want to put the two cards they reveal, to help alleviate that luck. That will increase draft time, but may be worth it.
Small World Draft is our first bidding format. Shuffle up your packs and deal five cards face up in a row next to the deck. Each player starts with 5 or so tokens and takes turns selecting a single card to take from the row. The furthest card is free. The second cards costs 1 and you pay by putting one of your tokens on the first card. When someone takes a card, they also get any tokens on it. The third card costs 2, which you pay by putting a token on the second card as well as the first. And so on. Whenever you buy a card, slide the cards behind it outward and put a new card from the deck into the last slot.
This takes a lot of time but is very cool. Do you bid the maximum to take a card when it first appears, or wait until your next turn to get it for 2 less? When do you take a weak card just to refill your token supply? Is that second card better than the first enough for you to put a token on the first, knowing your opponent will get the card and the token?
(I named this after Small World because it's better known and still available, but it's a remake of Vinci where I believe this bidding mechanic debuted.)
Auction Magic is a bidding format that happens while you play, not before. Invented by Reuben Covington and Dan Felder of Remaking Magic, I love quite a bit about this (but my lunch Magic partner doesn't). You shuffle up your packs into a single deck, and on each player's draw step, reveal one. You and your opponent bid on it, and whoever wins plays it immediately for free (or saves it to play for free later).
Shortcutting the mana economy of Magic changes the worth of cards drastically, as does the fact that you're in the middle of a game and so might be willing to pay more to get a blocker just to stem the blood flow from your opponent's turn one Moss Kami.
You're paying the tokens you bid to your opponent, so giving them even just 1 more than you means they can guarantee access to a huge bomb when it appears. Of course, if they do pay everything for that Blightsteel Colossus, you'll only have to bid 1 for that potential Angelic Edict, and you'll still have a lot more tokens than them. It's pretty sweet.
Epic is a Magic-inspired game that plays similarly to Auction Magic, minus the bidding. It's fast and smashy. It's not Magic, but it scratches the Limited Magic itch well and with zero set-up. Worth checking out.
Two-Headed In-Fighting is a Sealed format for two players. We improvised this format out of the frustration that the drafts we were doing weren't doing justice to the card pools. By collaborating to build two decks as if you were a two-headed giant team (from 8-12 packs), the decks that result will be as good as you can make with your shared pool.
We did this once, and the decks were vastly to superior to anything we'd drafted against each other. The bad news is, we failed to balance them and the blue-black deck won every game. The good news is, the games were all really good and pretty close. The white-green deck put up a good fight.
Color Draft replaces Arena Draft (check the comments to see why). It's deadly fast and offers the highest deck-quality-to-boosters-opened ratio after Two-Headed In-Fighting. Open six boosters. Sort them into five piles by color identity (Narnam Cobra goes in the green pile) and leave the colorless and gold cards unsorted. Randomly choose a first player. That player chooses an entire color pile or any single card (usually, one of the colorless cards). The other player chooses two. The first player chooses two, and so on, until all cards are taken. As soon as a player takes all the piles associated with a gold card's color identity, they get that card too. Now build and play.
Color Draft is efficient and satisfying. I recommend it highly.