If you're reading this then you're in an unfortunate situation! Here's some I've encountered and their solutions!
Oh no! My Worldgorger or Animate Dead is going to be destroyed!
Unfortunately this is the end of the line for you, all your permanents will be exile and there is no stopping it. To bounce back, hope you draw into 2 lands, Merchant of the Vale and another Animate Dead. This situation is the most dire of the bunch.
Oh no! I control a Gilded Drake!
This is fine, you may think you're in a corner, but you're really not. When Gorger exiles and returns all your permanents Gilded Drake will enter the battlefield under it's owner's control and exchange with another creature before Gorger's enter the battlefield effect will resolve. If you're goal was to rummage loop with Anje you're out of that line, but still get to loop Worldgorger. If your opponent tries to take Worldgorger with Gilded Drake then that's fine, you still control the Animate Dead and the combo will continue as normal because Worldgorger doesn't care who controls it as long as it exiles the Animate Dead. Gorger will exile your permanents and the Animate Dead's leave the battlefield effect will go onto the stack, causing Gorger's controller to sacrifice it. Gorger remembers the permanents it exiled regardless of who controls the trigger, meaning your permanents will enter the battlefield again bringing you back to the start of the loop with Gilded Drake's enter the battlefield effect on the stack.
Oh no! I can't loop a damage spell with the combo!
Your out is to loop any instant or sorcery with madness. Read each card as you draw them and see if it is relevant to the board-state. Looping cards like Distemper of the Blood, and Malevolent Whispers, or From Under the Floorboards can create an infinitely large zombie army, or steal every creature and make them all infinitely large with haste** . Just because you can't dome people with Fiery Temper doesn't mean you can't beat their faces in.
**Important note: To actually do this you need to have Merchant of the Vale on the field at some point along with infinite mana. This allows us to loop cards without dragon resetting the board, meaning you can make that token army without losing it to a worldgorger etb!
Oh no! I control lands that only deal me damage so I can't go infinite!
This is a bit of a multi-part scenario, although unlikely (In testing I've only encountered this once), it is possible to only control mana sources that deal you damage, causing you to not be able to execute certain loops or Avacyn's Judgment the win. An example being only controlling a Peat Bog and Mana Confluence. Chances are anyone playing this deck will never run into this issue, but it is worth pointing out. All of these scenarios assume that you control Anje and are not relying on Merchant of the Vale or Bag of Holding to rummage through your deck.
A sort of latest development from the Rakdos color discord is that you can loop Dark Ritual aswell as any other non madness card in your deck with Call to the Netherworld. Casting a non madness spell puts you down a card during the combo, but with CotN you net 1 card from casting it (you get a black madness creature back). That means aslong as you can cast ritual once (or put city of brass's trigger on the stack) you can filter your way into CotN and grab a black creature plus the draw from the looting then cycle Ulamog to reset your board state but with +3 black mana!
I control only a Tarnished Citadel!
This one is the easiest of them all. Your out is playing Avacyn's Judgment as you can still make infinite colorless mana with citadel with the added fact that judgement only requires 1 red to cast.
If Avacyn's Judgment is exiled, Alms of the Vein will net 0 life on each cast.
If both Alms and Judgment are exiled, From Under the Floorboards is your only hope.
I control only a City of Brass and/or Mana Confluence!
Alms of the Vein gains you more life than you lose per land activation.
Alternatively, your life total is the amount of times you can cast Fiery Temper. A sneaky trick with City of Brass is to activate before you discard Firey Temper so that you can respond to the damage trigger with Anje's discard ability.
Other than the above solutions, there isn't a clean way to win or put yourself in a favorable position.
Oh no! I think I'm Slow Playing!
First, thanks to Flooremoji for brining this up as it is pretty important.
Slow play if executing a loop is defined by the starting and ending board-states of that loop. The infamous 4-horsemen legacy deck loops Basalt Monolith and Mesmeric Orb to attempt to mill the top cards in a specific order to execute a win. The problem with that is if you don't mill the cards correctly, Ulamog, the infinte gyre will shuffle your graveyard into your library, returning you to the exact board-state that you started the loop at. The game only sees libraries a randomized, it doesn't actually know the order unless shown through scrying or whatnot.
For simplicity, looping madness spells with Ulamog, the Infinite Gyre does not count as slow play because you're never returning to your exact starting board-state. As long as the game has progressed between the start and end of your loop you're not at risk for slow play.
Now it's a different story if you only have 1 card in hand while you're rummaging. Because you cannot definitely claim that you will draw and cast a madness card before you draw Ulamog, this will be considered slow play. Because you have to discard your only card, you cannot hold Ulamog in hand and rummage to a madness spell, Ulamog will get discarded and shuffled, presenting the possibility of slow play.
Fixing Slow Play!
To fix a possible slow play situation, you must either net 1 card off of Call to the Netherworld so that you can keep Ulamog in hand and continue the loop or risk drawing a castable madness card off the top of your library before you draw Ulamog. Either way, play out the game. If you get hit with slow play, it's more than likely you were just unlucky and that's okay.