There are many indefinite loops possible in this game. The rule is that if the game ends up in an unstoppable loop, then it ends in a draw; the most common of these involves Animate Dead and Worldgorger Dragon. If it is stoppable, then the players simply decide how many times the loop occurs.
The usual trick with Animate Dead and Worldgorger Dragon is to combine it with another effect which can take place while one of the infinite looping abilities is on the stack, usually Bazaar of Baghdad to fill up the graveyard so Animate Dead can get a 20/20 with haste and flying or something similar.
The rules have on occasion been changed (used as a tournament rule when a draw is not an option) so that an unbreakable loop counts as a loss for the player who created it.
Magic has a bunch of cards with the Nightmare creature type. The most (in)famous of these is called Faceless Butcher. What this does is that when it comes into play, it removes a creature from the game other than itself. When it leaves play, the removed creature comes back. So, to draw the game, make sure there are no creatures in play. You need three Butchers (let's call them A, B and C.) Play Butcher A. Nothing happens since there are no legal targets for his first ability. Play Butcher B. B has a legal target: A. Remove A from the game. Play Butcher C. Butcher C has a legal target: Butcher B. Now B has left play, so the second ability triggers and resolves: Return the removed creature to play. The removed creature was Butcher A. A comes into play and has a legal target for its ability: C. Remove C from the game, which bring back B, which removes A... unless someone can either counter one of the abilities (only two or so cards in the entirety of the card pool targets triggered abilities) or can kill one of the butchers before the abilities happen, you've created an infinite loop and the game is a draw.
The Lorwyn/Shadowmoor blocks had several creatures with the Champion ability, which removes a creature you control from the game, often with restrictions on the sort of creature it can target. This can be used to create loops.
Also note that such loops can lead to a game-ending condition if combined with other cards like Pandemonium.
These days in Magic, the O-Ring Lock is better known than Faceless Butcher, with three of Oblivion Ring. Works exactly the same way, though, so if there are no other non-land permanents, you've just locked the game.
Someone got the situation semi-deliberately in an actual game in MtG Online. The server was not programmed to handle the situation and it kept crashing and restoring the game up to that point in a loop... ("Semi-deliberately" meaning that the situation came naturally in an actual game, and the player saw it coming and could have avoided it, but decided to cause the loop just to see what would happen.)
Play a Stuffy Doll, but target yourself with its damage-sharing ability instead of an opponent. Then enchant it with Pariah, which bounces damage off of you and back on to the Doll. Then tap it to deal one damage to itself. You now have one damage being passed around an infinite number of times. If they disenchant the Pariah, you take 1 from stuffy and that's it. (The actual process is. Stuffy deals 1 damage to itself, that ability resolves (stack is empty at this point), then Stuffy's ability is triggered, dealing 1 damage to you. Then when it begins resolving, pariah redirects the damage to itself. Then it triggers again. The only time to interrupt this loop is when the 1 damage goes on the stack, in which then pariah is no longer around when the 1 damage resolves, so it will be dealt to you.
If you have Repercussion, boom! You now have an infinitely increasing stack of damage constantly bouncing between you and the Stuffy Doll, with no way for it to ever resolve completely. Unless your opponent is sitting on a Disenchant, in which case he just waits until the damage is up around 10 billion or so, and then disenchants the Pariah...
Opalescence is an enchantment that turns all other enchantments into creatures. Day of the Dragons is an enchantment that exiles all of your creatures and replaces them with dragons until DotD leaves the battlefield. If you cast Day of the Dragons while Opalescence is on the battlefield, DotD will replace all of your creatures with dragons, including itself. but since that causes DotD to leave the battlefield, your newly created dragon tokens will go away and your original creatures will return, including Day of the Dragons. Dot D entering the battlefield again will cause all of your creatures, including Dot D, to be replaced with dragons, and so on, and so on...
Put both Platinum Angel and Transcendence on the battlefield. If you have (or reach) 20 life, transcendence will trigger its "lose the game" ability, which will do nothing due to Platinum Angel. However, after the ability resolves, it will automatically trigger again and do nothing. After that, it will trigger again. And again. And again...
Because of Platinum Angel (ability: you can't lose the game and your opponents can't win the game) and Abyssal Persecutor (ability: you can't win the game and your opponents can't lose the game) it is possible for two players to end up with some combination that prevents either from winning and then exhaust all possible means in either deck for the responsible cards to be destroyed. Not even a deck-out can end the game at that point.
In the old days, you were technically allowed to retreat a Pokemon as much as you wanted. What does this mean when you have two free retreaters? Simple: a never-ending game!
A particularly famous - albeit rare - example in the Pokemon TCG involves two primary cards to establish a perfect stalemate: Mewtwo LV.X (Legends Awakened), a Pokemon protected entirely from non-evolved Pokemon; and Uxie (Legends Awakened), a card able to return itself - and all cards attached - back to the deck via its Psychic Restore attack. So, when both players are using decks with both cards, as well as no evolved Pokemon, the game often ends perfectly tied, with no remedy per the rules in sight.
To make matters worse, this stalemate has no practical remedy in tournament play at all: if it happens, you're in for a long, drawn-out 40 minute round. When it's all over, the judges will either A) make you go to sudden death all over again, where this could repeat indefinitely, or B) simply give you and your opponent double game losses for delaying the event (ties are not allowed).