In the first chapter of The Hobbit, when Gandalf is talking to Bilbo, has Bilbo asked to go on an adventure?

Bilbo gets excited about Gandalf’s previous adventures, and Gandalf interprets that as Bilbo asking to take him on an adventure.

Respectable hobbits do not go on adventures. The Shire frowns upon anything exciting, and doesn’t approve of Gandalf because he has taken a few especially adventurous hobbits out adventuring. The Baggins family does not go on adventures, but the Took side has. Gandalf is relying on the Took side of Bilbo to get him going.

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Bilbo gets excited about Gandalf’s previous adventures, and Gandalf interprets that as Bilbo asking to take him on an adventure.

Respectable hobbits do not go on adventures. The Shire frowns upon anything exciting, and doesn’t approve of Gandalf because he has taken a few especially adventurous hobbits out adventuring. The Baggins family does not go on adventures, but the Took side has. Gandalf is relying on the Took side of Bilbo to get him going.

In this exchange, Gandalf is being playful and creative in order to get Bilbo to go on the adventure with the dwarves.  He twists his words so that Bilbo gets very confused and frustrated. Basically, Gandalf takes advantage of the fact that Bilbo is trying very hard to be polite. He gets him all muddled.

“ …Indeed for your old grandfather Took's sake, and for the sake of poor Belladonna, I will give you what you asked for."

"I beg your pardon, I haven't asked for anything!"

"Yes, you have! Twice now. My pardon. I give it you. In fact I will go so far as to send you on this adventure.” (Ch. 1) 

When Gandalf says that Bilbo asked to go an adventure, he is taking a few liberties with the conversation. Bilbo did get a little excited to hear Gandalf’s name, because he has heard stories of some fun adventures. He starts to say that life used to be interesting. 

"Not the Gandalf who was responsible for so many quiet lads and lasses going off into the Blue for mad adventures. Anything from climbing trees to visiting Elves or sailing in ships, sailing to other shores! Bless me, life used to be quite inter- …” (Ch. 1)

Gandalf turns his words back on him, and tells him that he will give him what he asked for—an adventure. Bilbo never asked for it outright, but Gandalf thinks from his reaction to the wizard’s name that he will be ready for one. Gandalf won’t take no for an answer. He will figure out how to get Bilbo there one way or another.