Raajaa Bali as such is not the character of Raamaayan or Mahaabhaarat, but his name is mentioned at several places as an example.

There have been many kings and sages whose names are famous for their Daan (charity, alms), such as: 
(1) Maharshi Dadheechi - to give his bones to Indra to kill Vritraasur. He made Vajra out of those bones;
(2) Raajaa Shibi - to give his own flesh to save the life of a pigeon from a falcon;
(3) Raajaa Rantidev - who did not take any money from his royal treasury for himself or his family. Once he went hungry with his family for days together and when he got some food to eat Devtaa asked for it in some form or other and he gave it to them happily.
(4) Raajaa Harishchandra - who donated the whole Prithvi and went to Kaashee with his family. Since the whole wealth was a little short, he sold his family to a Vaishya to work as maidservant and himself to a Dom (caretaker of the cremation ground) to pay off the remaining money. Later his son died and his wife brought his dead body to the same cremation ground where he himself was working. Since she didn't have any money for his Kafan (last cloth to be worn after the death), she brought his body naked. Harishchandra refused to accept it on the basis of the rules of cremation ground. His wife had to tear her half cloth to cover his naked body which made her half naked.

But still there are two very great Daanee's (charity giver, or donors) names are taken as an example in Hindu religious history - they are Raajaa Bali and Daanveer Karn. Karn is known only as Mahaa Daanee Karn.

Raajaa Bali was in the lineage of Maharshi Kashya and Prajaapati Daksh's daughter Diti. Diti had two sons - Hiranyaaksh and Hiranyakashyap. HIranyakashyap had four sons among whom the youngest one was Prahlaad because of whom Naaraayan appeared as Nrasinh Bhagvaan and killed Hiranyakashyap. Prahlaad had the son named Virochan and Virochan's son was Raajaa Bali, and Bali's son was Baanaasur whose daughter Ooshaa was married to Krishn's grandson Aniruddh. Because of Raajaa Bali Vishnu appeared as Vaaman Bhagvaan. Later Raajaa Bali was given the status of Indra in coming 8th Manvantar.

Raajaa Bali was a very powerful Raakshas. Once he won Trilok, so all Devtaa went to Vishnu and told their sad story. Vishnu consoled them and assured them to get their Swarg Lok back to them. 

After winning the Trilok, he was doing a Yagya conducted by Daitya Guru Shukraachaarya Jee. On the other side Aditi (Devtaa's mother) was very sad seeing her children (Devtaa) out of Swarg Lok. So she also prayed Kashyap Jee to help her children. Kashyap Jee said - "Don't you worry at all. Bhagvaan Vishnu will incarnate in our house and everything will be all right."

Bhagvaan Vishnu incarnated as Vaaman Bhagvaan - in a form of a short Braahman boy, only 52 finger tall (that is why He was called Baavan - Hindi word for 52) and got ready to go to Raajaa Bali's Yagya. He wore Rudraaksh necklace, yellow Dhotee (loin cloth), He had water pot in one hand and a Dand (staff) in His other hand and started for the Yagya.

That day was the last day of Yagya, and many beggars were getting their desired things from Raajaa Bali and were blessing him. When the Yagya was to end Raajaa Bali was about to perform the concluding ceremonies to complete the Yagya. He asked his people - "Is there anybody else who wants anything?" People checked and said "No". So he started to performing the concluding ceremonies of that Yagya. The then a servant came and informed him about a Braahman boy standing at the door. Bhagvaan Vaaman had arrived at his door and expressed His desire to get something from Raajaa Bali himself. 

The gatekeepers saw the valor of the Braahman boy and kept looking at Him. On the request of the boy, they went to Raajaa Bali and told that a boy wanted something from him. Raajaa Bali said "Let Him come in." Concluding ceremonies were halted and the boy was called in. Every body got stunned to see Him. He was so graceful that people there were not able to look at Him properly. Raajaa Bali also kept looking at Him. He had never seen such a gracious Braahman boy before.

He respected Him, washed His feet, offered an Aasan to sit on, and asked Him - "O Braahman boy, Ask what do you want? You will get anything you wish for - money, clothes, gems, animals, land anything." The Braahman boy said - "I don't need all these things. I am a Sanyaasee Braahman boy, what I will do of these things. If you really want to give me anything I want only my three feet land for my worship."

Raajaa Bali laughed and said - "O Braahman boy, Do you know who are you asking this donation from? I am Raajaa Bali. I can give you so much that even your ten generations will not be able to finish it. And you are asking me only your three feet land? Ask something substantial so that I can also feel honor in giving it to you."

Vaaman Bhagvaan said - "I don't need all these things, and I don't need more than that also, I am satisfied only with my three feet land for my worship. I am a Brahmchaaree Braahman boy so I will not have ten generations also. So, please if you can give me, give me only my three feet land."

Raajaa Bali laughed again. He again tried to convince Him that He could ask more but Vaaman Bhagvaan was adamant only on His three feet land. Then Bali said - "All right boy, be happy with your three feet land. What is three feet land for me measured by your such small feet." Saying this he took water in his hand and read the Sankalp (vow) for this donation. After that he asked Shukraachaarya Jee to do the proper rites for this donation.

The then Shukraachaary came to know that He was not an ordinary Braahman boy. He was Vishnu Himself and he had cheated Raajaa Bali by asking only three feet land. So he immediately said to Bali - "Hey Raajan, Stop reading Sankalp. He is not an ordinary Braahman, He is Vishnu Himself and He has come here to help Devtaa and to get Trilok back from you." 

Bali said - "Hey Guru, I have already started reading the Sankalp for this donation. I cannot take it back." But Guru Shukraachaarya Jee quoted several situations from scriptures when a Sankalp could be taken back. But Raajaa Bali didn't agree for it.

He said - "Otherwise also, nobody goes empty-handed from my door, therefore this Braahamn boy can also not go empty-handed from my house. If He has cheated me, that is His conscious. I have nothing to do with that. Thirdly, If He is Vishnu, then I am highly honored that even Vishnu who gives everything to everybody, today has come to my house to ask for something from me and I am able to give it to Him. This will be an example to the world that Bhagvaan Vishnu came to ask something from Raajaa Bali, and Raajaa Bali donated something to Bhagvaan Vishnu. So I cannot change my mind. You perform the rites for this donation."

When Raajaa Bali didn't listen to his Guru, he became furious. He gave Shaap to Bali that he would lose all his wealth soon, but Raajaa Bali didn't care for it. Shukraachaarya Jee also could not disobey Raajaa Bali, but when Bali was pouring water from the pot for the formal ceremony, he himself sat in the spout of that pot and stopped water to come out so that the ceremony could not be completed. Vishnu noticed it and saying this "What is inside it which is stopping water to come out?" He took a blade of Kush grass and inserted it in that spout. It pierced Shukraachaarya's one eye. Shukraachaarya Jee moved with pain and the water went out and the ceremony was completed. 

Now Raajaa Bali stood in front of that Braahman boy folding his hands and prayed Him and requested Him to measure His desired three feet land anywhere. Now the Braahman boy expanded His body so much that His head touched the sky. Raajaa Bali now got convinced that He was Vishnu in the form of the Braahman boy. He just stood admiring Bhagvaan still folding his hands.

Bhagvaan lifted His one foot and measured Bhoo Lok (Prithvi or Earth) in His first foot land, then He lifted His second foot and measured Swarg Lok in His second foot land and His body covered the sky. Now He asked Bali - "Where should I put my third foot to measure my third foot land?" 

Raajaa Bali had already made up his mind, he immediately lay down on his stomach [see Note below in red] and said to Bhagvaan - "My body is there for you to measure your third foot land. Please measure your third foot land here." Bhagvaan measured His third foot land there on his body.

Some scriptures say that he offered his head to Bhagvaan to put His third foot. And as Bhagvaan put His third foot on his head, he went inside the Earth up to Sutal Lok, so Bhagvaan gave him that place to live.

Bhagvaan blessed him and sent him to Sutal Lok to live because Bali lay down on the ground facing downward. Bhagvaan gave him the Vardaan that His Chakra (Divine disc) will always guard him. Prahlaad Jee also came there. Bhagvaan asked him also to go to Sutal Lok and live there and said - "You will daily see me there with my Gadaa and thus you will also be freed from your Karm bondage. Besides I will not kill anybody born in your lineage."

That is why when Raavan came to Raajaa Bali with the desire of the fight, he could not win him. Who could win Raajaa Bali whose doorkeeper was Vishnu Himself.

See   Raavan Meets Raajaa Bali

Baanaasur
Raajaa Bali had 100 sons. Among them Baanaasur was his eldest son. He was a great devotee of Shiv Jee and had 1,000 arms. One time when Shiv Jee was dancing His Taandav Dance, he played instruments with all his 1,000 hands. Shiv Jee got very pleased with his devotion and asked him to ask for any Var. He said - "You please live here to guard my city." Shiv Jee said - "So be it" Because of that Var Shiv Jee all Devtaa etc also used to serve him. 

Baanaasur was very proud of his 1,000 arms. Once he asked Shiv Jee - "I wish to fight with somebody, because my arms are itching for it." Shiv Jee said - "You will get opportunity to fight, do not worry." Later he had the opportunity to fight with Krishn. After the fight with Krishn in regard with his daughter Ooshaa, Krishn cut his all arms except four.