The Binding of Isaac: Rebirth

This challenge is easily one of the most difficult ones ever put into the game. You will be playing as Jacob and Esau, a character that consists of one "main" character, Jacob, and a secondary character, Esau, who is relatively weak. If either character dies, your run ends. The challenge consists of a Mom's Heart run, but the enemies in every room (including bosses) are doubled. As you can imagine, this can lead to some rooms being effectively impossible to clear without taking damage - especially as you have to dodge with two characters at once - unless you have some amazing form of room clear. While the duo is actually quite powerful, both equipped with 20/20 right off the bat, and able to enter all secret rooms throughout your run, this is not enough to carry you through the challenge on its own.

In this guide, I will be providing some tips and pointers that might make your attempts a bit smoother than mine were.

Tips: Tips and recommendations regarding the character.
Jacob and Esau: Playing Jacob and Esau is basically like playing co-op with yourself. Having two character at once provides a lot of benefits. When standing close to one another, their damage output is quite high. They also provide a number of other benefits, such as being able to hold different active items, being able to place 2 bombs for the price of one, and being able to pick up two rewards at the same time.
Bombs: As mentioned above, Jacob and Esau will always place one bob each, but always for the price of one. This changes how you have to think about your resource-economy, as it allows you break open a lot more of the environment at once. It also makes bomb synergies a lot more attractive than they already were.
HP: Given the amount of damage you will likely have to tank with both characters, getting your HP up is a good idea. Be very conscious of how you invest it if you decide to go for Deals with the Devil. Esau is especially vulnerable in the early game, and as such I recommend that players less confident in their ability to effectively control two characters at once try to keep him safe by feeding him HP while focusing on navigating the game with Jacob.
Builds: When deciding on how to distribute items between your characters, keep in mind that they are distinct, and that items (with a few exceptions) do not grant synergies between the two. It is entirely possible to transform one character, but not the other. If, for example, a Guppy transformation is achieved through passive items, only the character holding those passive items will be transformed. I strongly recommend that defensive items go toward keeping Esau safe, while offensive items be given to Jacob as the character you are in direct control of.
Flight: While flight is usually a very powerful upgrade, I do not recommend taking any flight upgrade on this challenge, unless you are in an extremely fortunate - and extrenely rare - situation where it is available to both characters (E.G. a Deal with the Devil offering you both Lord of the Pit and Spirit of the Night). This is because while one character flies, the other will still be traveling by foot, and will more easily get stuck on the environment or step on spikes, as they are become more difficult to control.
Navigation: Moving around with these two characters is challenging during the normal game, and having to do so with double the amount of enemies that you're used to isn't any easier. You will no doubt experience that one character gets stuck or behaves in an unusual manner. To keep things simple, I recommend keep the two characters as close together as possible during combat. This can be achieved by forcing them into a corner in case they get separated.
Bugs and issues: Note that given the double-spawns, not every room or enemy placement will work as intended. Some enemies may spawn inside walls or pieces of the environment, leading to them getting stuck. They may behave erratically or remain passive. They can usually be dislodged by simply shooting at them. Some rooms and bosses can also become effectively unwinnable, given the sheer amount of enemies you have to contend with. When or if that happens, the best piece of advice I can give you is to remain patient and try again.

Items: A short list of items that I found particularly useful.
Necronomicon: An active item that deals a significant amount of damage to everything inside the room that you're in. This is a relatively common item that can help you quickly decimate your opposition. I recommend using it sparingly, preferably on large rooms or rooms that have lots of weak enemies swarming you at once.
Skatole: A defensive item obtained from the Shell Game beggar, this item will pacify or slow down most types of flies that you encounter, making the rooms that contain them (and even a handful of bosses) significantly easier to deal with.
Anarchist's Cookbook (and The Tower): A very common item, this proved to be a bit of an unsung hero on my winning run. It should be used very sparingly. In fact, I only used it for bosses. It did, however, allow me to effectively one-shot all bosses leading up to the womb, and for that it has earned a permanent spot in my personal hall of fame. While the obvious dangers of using it might make it seem counterintuitive on a run that is already so difficult, keep in mind that even if you should end up tanking an explosion, it is often worth doing so rather than tanking several hits from a double-boss. I would never have gotten past an early game double champion Ragman and their spam-summoning if not for this.
Brimstone and technology: Lasers of all kinds (Including Shoop da Whoop!) are especially valuable when your main goal is to clear a room as fast as possible.
Crystal Ball: While I would normally argue that Satanic Bible, Book of Revelations and The Nail all present stronger defensive items than this, the fact that Crystal Ball spawns spirit hearts on the ground, rather than adding them to the character using it, proved invaluable to me. This allows you to pick it up with whichever character needs it the most, regardless of who holds the items.
Orbitals: Orbitals of all kinds are great for protecting Esau.
Battery upgrades: It goes without saying that if you're relying on an active item to carry you through this challenge, you will want to be able to use it. 9 Volt, Charged Baby, The Battery, Car Battery, and other charge-related items will help you achieve this, and can turn items like Crystal Ball or Sulfur may carry you through your run.
2Spooky: An item that causes all enemies that go near the holder to turn away in fear. This is a very powerful defensive item to have on you in rooms full of melee-oriented enemies, as it may effectively make you untouchable.
Bomb upgrades (And Mama Mega): As mentioned above, bomb synergies are particularly powerful. Every 5 bombs you gain from picking one up effectively becomes 10, increasing their flat value, but a variety of effects may also help you clear rooms faster or more safely, such as Brimstone Bombs or Butt Bombs.

Hot Potato

An exceptionally difficult challenge, Hot Potato sees you playing as a version of the Tainted Forgotten whose body explodes every 2.5 seconds. This will be your only mode of attack throughout the challenge. All item interactions will also, if possible, prefer your body over your soul. What his means is that any character upgrades, such orbitals, will go to your body. Your body will pick up items it comes into contact with, including items in special rooms, such as shops. Dropping items or bombs will cause them to drop where your body is located. Items or consumables spawned by cards or items will spawn where you body is located. Most importantly, the explosion will also affect you, as well as your enemies. Your goal is to beat Satan.

Tips: Tips and recommendations regarding the character.
Bomb Upgrades: Certain bomb upgrades will apply to your body, such as Brimstone Bombs. These may prove immensely helpful in clearing rooms. Some may recommend playing Bomb Beggars to obtain bomb upgrades, but in my experience these very rarely pay out, and I thus cannot recommend this. I found the bombs were more useful when dropped just prior to your body exploding, in the hopes of them being knocked into bosses. This was especially true when a bomb effect had been obtained.
Explosion Immunities: Host Hat and Pyromaniac are absolutely invaluable. However, it goes without saying that actually obtaining either of these is exceedingly rare and not a reliable strategy. Should either be obtained, this will allow you to basically hold your body over you head and tank the explosions, while also making the most difficult fights of the run (Mom and Satan) significantly easier.
Health: Your only source of health will be Soul Hearts, Black Hearts, empty Bone Hearts and Eternal Hearts. As such, any items or trinkets that may grant you more of these, such a Book of Revelations or Mom's Pearl, are exceedingly valuable. I do not recommend Satanic Bible for this challenge, as your inability to generate heart containers makes Devil Deals significantly less attractive.
Contact Damage: Orbitals, The Virus and other items that are most effective when making contact with an enemy are strongly recommended, as they will make the act of throwing your body at enemies and bosses significantly more impactful. The impact damage at the beginning of the challenge is extremely lackluster.
Flight: Obtaining any form of flight will cancel the bouncing effect of your body on exploding, It will instead become a static entity, staying in place until picked up or otherwise forced out of the way by an outside influence. I personally found this to be extremely useful.
Patience and Rocks: Given the unlimited amount of explosions, it can pay off to clear all or most of the environment. Pots and skulls may yield money or health, and rocks may reveal crawlspaces that, in turn, may yield valuable items.
Skips: Ehwaz runes and similar effects will allow you to bypass difficult bossfights. I recommend using these, if available, to skip Mom and/or Mom's heart, as these are in my experience by far the most difficult encounters prior to Satan.

Strategy: My guide on how to overcome this challenge.
Magic Fingers: Clearing rooms and killing bosses is extremely difficult when all you have is a big bomb you can throw around. Obtaining an alternative means of dealing damage to foes is thus extremely valuable. Necronomicon or Anarchist's Cookbook may be useful for this purpose. Dry Baby is also a very valuable pickup. As are bomb upgrades, as outlined above. However, the most effective and reliable strategy I found, which can be executed semi-consistently, is to obtain Magic Fingers from a shop, while making effective use of your unlimited explosions and flight to obtain as many coins as possible, and using these where necessary to clear rooms that are otherwise difficult to manage. It may also be spammed to quickly dispatch bosses. Note that Magic Fingers has been buffed for Repentance, and now deals very significant damage. Given your high starting damage, and the effect of further upgrades or temporary stat boosts, this item may give you the edge you need, as it did for me. Defeating Satan from phase 2 and onwards with around 7 base damage cost roughly between 40-50 coins, with the rate at which you dispatch him determined by your skills at button mashing.

Defeat Satan as Isaac.

Defeat Loki.

Beat the challenge "When life gives you lemons"

Defeat Mom's heart 5 times.