Producing Graham's number means do anything that requires Graham's number to represent the game state. For example, you could produce Graham's number 2/2 zombie tokens, have Graham's number life, or have a creature with Graham's number in power and/or toughness.
You cannot use a combo that can be repeated indefinitely (commonly referred to as a "infinite combo"), as defined in CR 726. For example, you can't use Basalt Monolith enchanted by Power Artifact, or Squirrel Nest in combination with Earthcraft as either of these can be repeated an arbitrary number of times. You can use abilities repeatedly, such as the ability on Staff of Domination or Voltaic Construct, as long as there is some in-game limit on the number of times it can be repeated (other than the the rules requirement that you not repeat an optional loop forever). The key text from CR 726 (describing interactions that this challenge disallows) is the following:
726.1b Occasionally the game gets into a state in which a set of actions could be repeated indefinitely (thus creating a “loop”). In that case, the shortcut rules can be used to determine how many times those actions are repeated without having to actually perform them, and how the loop is broken.
If the combination involves any cards that let players "choose a number", such as Menacing Ogre, numbers greater than 256 cannot be chosen.
You must get to Graham's number in at most 256 turns. (Note that taking infinite extra turns is prohibited as that would be an infinite combo.)
You have 256 mana of any color available to you on each turn without it counting against your card limit. Mana beyond that requires you to add cards to do this.
You have a single opponent with 20 life. The opponent takes no actions, and does not lose from drawing from an empty deck (assume that they have a deck made of 300 basic lands). You cannot rely on the opponent making specific choices unless you have an effect that lets you control their choices.
You must start with a legacy legal deck. In particular, all cards used must be legal in the Legacy format. Additionally, standard deck building rules are in effect; i.e. no more than four copies of any card can be used unless that card specifies otherwise.
All of your cards start in either your hand or library (your choice which for any given card). To get a card into another zone requires an ability that moves it there.
The solution that uses the fewest number of cards from it's deck and sideboard to complete this task wins. Using a card means that having that card in your deck or sideboard is necessary to completing the combo. Playing the card is the most common "use" of a card, but using a card also includes using it's activated or triggered abilities from a zone outside of the battlefield (such as Decree of Savagery or Bridge from Below) If you use multiple copies of a card, each copy counts as a separate card. Cards that are in the deck merely to meet the 60 card minimum but otherwise are never used are not counted.
Inspired by: https://boardgames.stackexchange.com/q/57658/9999
For those who haven't played Magic the Gathering extensively, you don't need to know the entire rule set (which is quite long), as most of that deals with interactions between players, which isn't present for this challenge. The basic rules, supplemented by the comprehensive rules on spells and abilities (section 6) is probably most of what you need to know. The harder part is finding cards that are helpful (as Magic has over 20,000 cards that have been printed). This is the best card search I know about: https://scryfall.com/advanced. Here are some examples of helpful searches: