There are lots of stories regarding this, I am listing some of the common ones:

Humans:
1. Bhagirath- He was the descendent of the great King Sagara; born without bones and was not able to walk properly. Once Astavakra muni saw Bhagirath and thought he was mimicking him. He got angry and said if its true then he should really become like that,else he would become healthy. And thus the boy became healthy!! Later Bhagirath mediated and pleased Brahma to bring Ganga from heaven to give moksha to his ancestors. To stop Ganga's heavy flow, he had to pray Shiva who agreed for the same. But Ganga being arrogant thought of sweeping Shiva and thus to teach her a lesson, Shiva kept her locked in His jataa/hairlocks. Bhagirath had to please Shiva again and let Ganga go free.

2. Nara and Narayana - The twins sons of Dharma (two incarnations of Vishnu), performed severe penance in Badrikashraya of Bharat Khand, in front of a Shivalingam fashioned out of earth. Pleased with their devotion, Lord Shiva appeared in front of them and said that they may ask for a boon. Nar and Narayan requested Shiva to take up a permanent abode as a Jyotirlingam at Kedarnath so that all people who worship Shiva shall be freed from their miseries. Also, since both of them performed great penances at the holy spot of Badrinath, in Badrinath Temple's sanctorium, to the far right side of the stone image of Badri-Vishala (or Badri-Narayana), are the images of Nara and Narayana.

3. Arjuna - The 3rd pandava meditated and pleased Shiva to attain the divine weapon, Pashupatastra.

4. Bhisma- Although he didnt do any penance, but Bhisma was granted the boon of Sweccha Mrityu (control over his own death — he could choose the time of his death) from his father, Shantanu. So, after getting defeated by Arjun, he laid on a bed of arrows and delayed his death by his own will.

Asuras/Demons:
1. Hiranyakashipu- (Already mentioned in one of the answers in elaborately); Was finally killed by Lord Vishnu as Narasimha.

2. Holika - She was sister of Hiranyakashipu and had a special boon from Brahma that prevented her from being harmed by fire. Hiranyakashipu attempts to kill his son Prahlada by making him sit in burning pyre with his sister Holika. Prahlada chanted Vishnu's name and in the battle of good against evil, Holika was burnt down but nothing happened to Prahlada.

3. Putna - (though this is not the correct answer to the question asked as the lady didnt do any meditation, but still included due to the boon she got) - In her previous birth, she was Ratnamala, daughter of king "Bali" (grandson of Prahalad). When Vishnu appeared as Vamana in the yajna performed by Bali, she was so attracted by Him that she wanted Him as a son so as to feed Him. But after Vamana sent her father to patal, she got angry and desired if she would be having a son like Him, she would give Him posion. Lord Vishnu granted her both the boons. So in dwapar when Kansa sends Putna to kill Krishna, she tries to kill Him by breastfeeding.

4. Mahisasura - performed severe penance to please Brahma and finally obtained a boon from him to the effect that he could neither be killed by Gods nor by man or animal, leaving women in his list. Finally got killed by Goddess Durga.

5. Shumbha Nishumbha - got boon from Brahma to be killed by a woman. They were killed by Devi Durga. Other demons like Chand Munda and Raktabeeja were also killed by Devi Durga.

6. Ravana, Kumbhakarna and Bibhishan, all the three brothers performed severe penance to please Brahma.
During his penance, Ravana chopped off his head 10 times as a sacrifice to appease him. Each time he sliced his head off a new head arose, thus enabling him to continue his penance. At last, Brahma, pleased with his austerity, appeared after his 10th decapitation and offered him a boon. Ravana asked for immortality, which Brahma refused to give, but gave him the celestial nectar of immortality. The nectar of immortality, stored under his navel, dictated that he could not be vanquished for as long as it lasted.Ravana also asked for absolute invulnerability from and supremacy over gods, heavenly spirits, other rakshas, serpents, and wild beasts. Contemptuous of mortal men, he did not ask for protection from these. Brahma granted him these boons in addition to his 10 severed heads and great strength by way of knowledge of divine weapons and magic. Vishnu appeared as Ram, an ordinary human being and killed Ravana.

Kumbhakarna wanted "Indraasana" (seat of Indra) but on the request of devtas, Devi Saraswati sat on his tongue and made it " Nidraasana" (bed for sleeping). It is also said he intended to ask for Nirdevatvam (annihilation of the Devas), and instead asked for Nidravatvam (sleep). On the request of Ravana and Bibhisan,Brahma mitigated the power of the boon by making Kumbhakarna sleep for six months and being awake for rest six months of a year.

Bibhishan although he was born in the demon race, he was alert and pious and got boon from Brahma in which he would always be at the feet of the Lord.

Source: Grandmaa and wiki ofcourse :)