I realize that some of you might be stopping in because of A-Z. You might not have played D&D and you may not have any interest in it. However, if you do you can obtain the rules for the current edition online for free (though its' not all the material in the core rulebooks, it's all you need to run an extended game). All you need is this and a set of polyhedral dice and you're good to go. I'll also include some free supplements and things online.
There are those that argue it's not a complete game. I disagree. This is everything you need to run a game from level 1-20 (minus dice).
I'm keeping this post prominent, even after the challenge is over. It was last updated: 4/26/17
Basic Rules for Dungeons and Dragons
5E SRD - This is the no frills system reference document for 5th edition. It contains much more material than Basic, though not quite as much as the core rulebooks.
Hoard of the Dragon Queen Supplement - new magic items, monsters, and spells
Rise of Tiamat Supplement - new magic items, monsters, and spells
Elemental Evil Player's Companion - new races and spells
Curse of Strahd Character Options - new background and gothic trinkets list
Gray Dwarf Subrace - new subrace for dwarves, usable with basic
Kits of Old - 2 new builds for bard (unusable without core book), 2 fighter archetypes (usable with Basic)
Urban Bounty Hunter Background - new background, usable with basic
Eberron - new races, wizard tradition, and feat (mostly usable with Basic)
Ranger - not usable with Basic
Variant Rules - could be usable with Basic
Mass Combat Rules - usable with Basic
Modifying Classes - given the limited nature of Basic, this is quite useful
Mastermind Archetype - new rogue archetype, usable with Basic
Waterborn Adventurers - new race and archetypes (mostly usable with Basic)
Modern Magic - usable with Basic
Rune Magic - presents the first 5e prestige class and discusses prestige class construction, usable with Basic
Light, Dark, Underdark - this contains several new class options, though only one of them is useful with Basic.
The Mystic Class- This is a 20 level base class usable with Basic
That Old Black Magic - This contains some material usable by with Basic (notably the Abyssal Tiefling race)
Gothic Heroes - This contains a new subrace and two new archetypes, all of it is usable with Basic
Plane Shift: Zendikar - This setting is fully compatible with basic. It has several new races and a few new monsters (all inspired by Zendikar setting of Magic: The Gathering).
Feats - New feats (and the thinking behind them). This is all compatible with Basic.
Plane Shift: Innistrad - This setting is fully compatible with basic. It has four human subraces (though they don't call them that, a new background, and a several new monsters and monster abilities (all inspired by the Innistrad setting of Magic: The Gathering)
The Faithful - While the warlock stuff can't be used with Basic, Arcane Tradition: Theurgy works wonderfully with Basic
Barbarian: Primal Paths - New primal paths for barbarians. Not compatible with Basic.
Bard: Bard Colleges - Two new bard colleges. Not usable with Basic.
Cleric: Divine Domains - Three new cleric domains. Usable with Basic.
Fighter: Martial Archetypes - Four new fighter archetypes. Usable with Basic.
Druid: Druid Circles and New Wildshape Options - New options for druid characters. Not usable with Basic.
Paladin: Sacred Oaths - Two new sacred oaths for paladins. Not usable with Basic.
Ranger and Rogue: New Archetypes - The scout archetype can be used with Basic.
The Artificer - This new class can mostly be used with Basic.
Plane Shift: Kaladesh - This setting is fully compatible with basic. It has several new races and a few new monsters (all inspired by Kaladesh setting of Magic: The Gathering).
Warlock and Wizards - Two new Warlock patrons and a new Arcane Tradition (which can be used with Basic)
Wizard: New Arcane Traditions - Two new arcane traditions for Wizards (compatible with basic)
Sorcerers - Three new sorcerous origins (not compatible with Basic)
Traps Revisited - New traps and trap creation (compatible with Basic)
A Trio of Subclasses - options for paladins, rangers, and monks (not compatible with Basic)
Mass Combat - Rules for mass combat encounters (compatible with Basic)
Starter Spells - New starting spells for all classes (some are compatible with Basic)
Feats for Races - Some of these feats can be used with Basic.
Feats for Skills - Some of these feats can be used with Basic