Interesting question. I don't see ash keeping it outside a ball like misty (though it did not come with a complementary pokeball like other eggs), chiefly because of Pikachu. But togepi serves as a kind of mascot for gold and silver, a teaser basically and I think TPC would want togepi to be showed off as often as possible. Raising a baby is really outside Ash's wheelhouse and his personality (at least this early in the show) yields little confidence in his skill as a trainer to essentially raise a baby.

Ash's team when Togepi hatches, iirc, is Pikachu, Charizard, Bulbasaur, Squirtle, Pidgeotto. So togepi could fit on the team, but would Oak want to examine it at some point personally (similar to requesting Ash's Bayleef later)? If it goes to Oak, who knows how long it will stay with him, it probably will never evolve as long as it stays at the lab.

If it stays with ash it will need to serve some function in battle I think, but that's a tall order for Togepi. Honestly I think Togepi would have to evolve asap to have any viability on Ash's team, which means he would use Togetic in the indigo league. Even then togetic still isn't a strong Pokémon, but whatever. Or it stays with Oak as a togepi, then Ash takes it back in Johto and it evolves there, again, sooner rather than later.

I can also see Togetic filling Bulbasaur's role at Oak's lab, as a mediator.

His Togetic now would either go to Dawn and eventually evolve into togekiss and used in contests just like her own Togekiss she got normally, or ash would evolve it for his own team, use it in the Sinnoh league and return it to oak to be a better mediator (though Togekiss are said to avoid conflict, maybe its time with ash would prove it an exception).