Yes. FREE POKEMON! Make sure you have room in your party to pick up these Pokemon.
This isn't technically a gift Pokemon but you could still keep it for yourself. If you go to the gate north of Goldenrod City, a man in red clothes will ask you to bring his Spearow with mail attached to it to a friend on Route 31. His friend is one of the trainers you probably battled already. If you bring the Pokemon to the friend, you will give the Pokemon to the friend and get a TM in return.
When you went to Ecruteak City, you bumped into a guy named Bill [No, not Bill Gates] in the Pokemon Center. Bill was apparently the one who came up with the Pokemon Storage system with his friends. He went back home after he met you. To get a Pokemon from him, go to Goldenrod City. Search for him in one of the houses [To the West] and talk to him. He'll offer you an Eevee. Accept and it is all yours.
In one of the South-West houses of Cianwood City, a guy named Kirk is worried about his Pokemon. He is afraid that somebody will steal his Shuckle. Although you are a complete stranger, he'll offer you to look after it. It holds a Berry Juice.
In Mt.Mortar, the Karate King is training with his Pokemon. When you talk to him, he will battle with you. He has Fighting Pokemon [I believe Hitmontop, Hitmonlee, & Hitmonchop] and after you beat him, he will give you a Tyrogue.
After you have answered the questions at the Dragon's Den, you should go back near the entrance, but Clair is in the way. She will give you a TM and run off. Go back to the Elders and talk to the one who asked the questions. Depending on how "good" you answered his questions, he will give you a Dratini with a special move, Extreme Speed. If you didn't, then you get a basic Dratini.
After getting all 16 badges and defeating Red at Mt.Silver, head down to Pallet Town. Go inside Professor Oak's Lab. He will compliment for your success and will let you pick one of the 3 Kanto Starters for yours to keep.
After defeating Red and getting all 16 badges, head on over to Saffron City. Enter the Silph Co. Building. Steven Stone, Champion of the Hoenn Region, will let you pick a Stone. You have the choices of the Red Stone [Torchic], Green Stone [Treecko], and the Blue Stone [Mudkip]. Pick whatever stone [Pokemon] you want and he will give you that Pokemon.
- If you were wondering how to get to the Karate King at Mt.Mortar, check out this web page to see a map of the area.
- Togepi is a Gift Pokemon, but you would have to get it's egg to continue on in the story so it is not listed here