Kakagai fic prompt #20
Out of deep respect and honor for Kakashi's beloved sensei, and with the understanding that the Hokage is concerned about Naruto's origins being revealed if Kakashi gets close to him of his own volition, Gai volunteers and is accepted (as a shinobi unaffiliated with any clan or the Yondaime Hokage) to take in and raise Naruto.
Variations thereupon:
Kakashi needs to be introduced to Naruto in between missions like a nervous pet meeting the new one; a little at a time, and allowed to run away when it becomes too overwhelming
Kakashi marks a date on the calendar upon which it will no longer be suspicious for him to propose to Gai so he can get in on the baby-raising goodness
Kakashi spends so much time out of the village on missions that he doesn't find out until Naruto is nearly three years old
Naruto has to decide how he feels when Gai also adopts Lee and/or Tenten at some point a few years down the line
Naruto is not the deadlast when he actually has someone(s) at home to care for him and becomes a taijutsu master, much to everyone's bafflement and (probable) dismay
Gai keeps showing up in the ANBU barracks with infant Naruto (he's not supposed to know where they are) (he insists it's fine because no one in the village will see Kakashi interacting with Naruto there therefore it would only lead to them being compromised if the Hokage didn't trust his own ANBU; you do trust your own ANBU not to turn traitor and reveal state secrets, don't you Hokage-sama?)
Naruto is the Jounin's collective baby
Naruto is the ANBU's collective baby, in spite of the fact that Gai is not himself a member of ANBU
Somehow this single event (Gai adopting Naruto) averts the Uchiha Massacre
Somehow this single event (Gai adopting Naruto) averts Hyuuga Hizashi's death
Within 24 hours of being granted custody over Naruto, Gai pulls some ancient clan bylaws to justify him and Kakashi getting married and Kakashi agrees solely to be in Naruto's life; they don't become romantic until many years have passed.
Within 24 hours of being granted custody over Naruto, Gai pulls some ancient clan bylaws to justify him and Kakashi getting married and Kakashi falls in love with him immediately, quitting ANBU to dedicate himself fully to raising Naruto. It takes Gai a while to figure out that Kakashi is 100% on board with being married and in love and not just in it for Naruto alone.
Gai tries to get Naruto a green jumpsuit but gives in when Naruto instead seems to fall in love with his orange leg warmers. Naruto gets an orange jumpsuit instead.
Kakashi, once emotionally stable enough, regularly steals Naruto to ensure Gai isn't corrupting him completely. The Hokage is completely unable to prove that Kakashi is doing so because no one is willing to come forward as a witness to said child thefts.
Naruto has a favorite turtle summon.
Naruto is very popular among the turtle summons.
Naruto signs the tortoise summoning contract instead of the Toads. Jiraiya has to deal with his feelings on this.
(all of the above can be mixed and matched as desired)