Some of the Sahaabah (may Allah be pleased with them) who memorised the Book of Allah, may He be exalted

Firstly:

The Companions of the Prophet (blessings and peace of Allah be upon him) were the most eager of people to do good and the swiftest in doing righteous deeds, and the most able to do such deeds.

One of the greatest of their concerns was memorising the Book of Allah, may He be exalted, learning it and acting upon it, then teaching it to people. They used to learn it from the Messenger of Allah (blessings and peace of Allah be upon him), and they would teach it to one another. So whoever missed out on learning something from the Messenger of Allah (blessings and peace of Allah be upon him), because they were preoccupied with jihad, would learn it from his companions.

The most famous of those who used to teach people the Quran include: ‘Uthmaan ibn ‘Affaan, ‘Ali ibn Abi Taalib, and Ubayy ibn Ka‘b (may Allah be pleased with them). They taught many of the younger Sahaabah and, after that, many of the Taabi‘een.

Abu ‘Abd ar-Rahmaan as-Sulami said: Those who used to teach us the Quran told us that they used to learn it from the Prophet (blessings and peace of Allah be upon him). When they learned ten verses, they would not move on until they put into practice what those verses said, so we used to learn the Quran and how to act upon its teachings together."(Tafseer al-Qurtubi  1/80).

So the Sahaabah (may Allah be pleased with them) strove to memorise the Book of Allah and teach it to people and help them to memorise it.

Suwayd ibn ‘Abd al-‘Azeez said: When Abu’d-Darda’ had prayed Fajr in the mosque of Damascus, the people would gather to learn Quran from him, so he would put them into groups of ten, with an instructor for each group, and he would stand in the mihrab watching them. If one of them made a mistake, he would refer to his instructor, and if the instructor made a mistake, he would refer to Abu’d-Darda’ and ask him about that.

It was narrated that Ibn Mashkam said: Abu’d-Darda’ said to me: Count those who are studying the Quran with me. I counted them, and they were over one thousand and six hundred, and for each group of ten there was a teacher."(Ma‘rifat al-Qurra’ al-Kibaar, p. 20)

Secondly:

A great number of the Sahaabah memorised the Book of Allah, may He be exalted, and they were so great in number that they could not all be listed. A number of the scholars mentioned the most prominent of those among them who memorised the Book of Allah.

Al-Haafiz Ibn Hajar (may Allah have mercy on him) said:

Abu ‘Ubayd mentioned those among the Companions of the Prophet (blessings and peace of Allah be upon him) who memorised the Quran. Among the Muhaajireen he listed the four caliphs, Talhah, Sa‘d, Ibn Mas‘ood, Hudhayfah, Saalim, Abu Hurayrah, ‘Abdullah ibn as-Saa’ib and the ‘Abdullahs. Among the women he listed ‘Aa’ishah, Hafsah and Umm Salamah. But some of these only completed its memorisation after the Prophet (blessings and peace of Allah be upon him) had died.

In Kitaab ash-Sharee‘ah, Ibn Abi Dawood listed among the Muhaajireen also: Tameem ibn Aws ad-Daari and ‘Uqbah ibn ‘Aamir. Among the Ansaar he listed: ‘Ubaadah ibn as-Saamit, Mu‘aadh who was known by the kunyah Abu Haleemah, Mujammi‘ ibn Haarithah, Fadaalah ibn ‘Ubayd, Maslamah ibn Makhlad, and others. He stated that some of them only memorised it after the Prophet (blessings and peace of Allah be upon him) had died. Among others who also memorised it was Abu Moosa al-Ash‘ari, who was mentioned by Abu ‘Amr ad-Daani. Some later scholars also listed ‘Amr ibn al-‘Aas, Sa‘d ibn ‘Abbaad and Umm Waraqah as being among those who memorised the Quran."(Fath al-Baari  9/52)

See also: al-Itqaan by as-Suyooti (1/248-249)

Among the most prominent of these Sahaabah (may Allah be pleased with them) are the following:

1. Abu Bakr as-Siddeeq (may Allah be pleased with him)

He died on a Monday in Jumaada al-Oola 13 AH at the age of sixty-three. ‘Umar led the funeral prayer for him, and he was buried with the Messenger of Allah (blessings and peace of Allah be upon him).

Tahdheeb at-Tahdheeb (5/316)

2. ‘Umar ibn al-Khattaab

He was killed on a Wednesday 26 Dhu’l-Hijjah 23 AH, at the age of sixty-three; there are other views concerning his age.

Tahdheeb at-Tahdheeb (7/441)

3. ‘Uthmaan ibn ‘Affaan (may Allah be pleased with him)

He was one of those who memorised the Quran at the time of the Messenger of Allah (blessings and peace of Allah be upon him). He died a martyr, unlawfully killed in his house, on 18 Dhu’l-Hijjah 35 AH, at the age of eighty-two, according to the correct view.

4. ‘Ali ibn Abi Taalib (may Allah be pleased with him)

He was killed in Kufah on the morning of 17 Ramadan 40 AH. He memorised the Quran after the death of the Prophet (blessings and peace of Allah be upon him), and it was said that he memorised it during his lifetime.

5. Ubayy ibn Ka‘b (may Allah be pleased with him)

He was the most prominent of the Quran-reciters, the one who had best memorised the Book of Allah. He memorised it at the time of the Messenger of Allah (blessings and peace of Allah be upon him) he died in Madinah in 20 AH or 19 AH.

6. ‘Abdullah ibn Mas‘ood ibn Ghaafil (may Allah be pleased with him)

He memorised the Quran at the time of the Messenger of Allah (blessings and peace of Allah be upon him). He died in Madinah at the end of 32 AH.

7. Zayd ibn Thaabit ibn ad-Dahhaak (may Allah be pleased with him)

He memorised the Quran at the time of the Messenger of Allah (blessings and peace of Allah be upon him). He died in 45 AH, according to the more correct view.

8. Abu Moosa al-Ash‘ari, ‘Abdullah ibn Qays (may Allah be pleased with him)

 He learned the Quran from the Prophet (blessings and peace of Allah be upon him). He died in Dhu’l-Hijjah 44 AH according to the correct view.

9. Abu’d-Darda’ ‘Uwaymir ibn Zayd al-Ansaari (may Allah be pleased with him)

He learned the Quran at the time of the Prophet (blessings and peace of Allah be upon him). He died in 33 AH.

10. Abu Hurayrah ad-Dawsi (may Allah be pleased with him)

He learned the Quran from Ubayy ibn Ka‘b. He died in 57 or 58 or 59 AH.

11. ‘Abdullah ibn ‘Abbaas ibn ‘Abd al-Muttalib (may Allah be pleased with him)

He learned the Quran from Ubayy ibn Ka‘b. He died in at-Taa’if in 68 AH.

12. ‘Abdullah ibn as-Saa’ib ibn Abi’s-Saa’ib, the most prominent reciter of the people of Makkah

He was a companion of the Prophet (blessings and peace of Allah be upon him) and narrated a few reports. He was one of the younger Sahaabah. He learned the Quran from Ubayy ibn Ka‘b.. He died around 70 AH, during the governorship of Ibn az-Zubayr.

See: Ma‘rifat al-Qurra’ al-Kibaar by al-Haafiz adh-Dhahabi (p. 10-25).

13. Mu‘aadh ibn Jabal (may Allah be pleased with him)

He learned the Quran from the Messenger of Allah (blessings and peace of Allah be upon him). He died in 17 or 18 AH, at the age of thirty-four.

Tahdheeb at-Tahdheeb (10/187).

14. Saalim the freed slave of Abu Hudhayfah (may Allah be pleased with him)

He learned the Quran from the Messenger of Allah (blessings and peace of Allah be upon him). He died in 11 AH on the day of al-Yamaamah.

Al-Bidaayah wa’n-Nihaayah (9/497).

15. ‘Abdullah ibn ‘Umar ibn al-Khattaab

He learned the Quran from the Prophet (blessings and peace of Allah be upon him) and from ‘Uthmaan ibn ‘Affaan and Zayd ibn Thaabit. He died in 73 AH or, it was said, in 74 AH.

Tahdheeb at-Tahdheeb (5/330)

16. ‘Abdullah ibn ‘Amr ibn al-‘Aas (may Allah be pleased with him)

Ibn Maajah (1346) narrated that he said: I memorized the Quran and once I recited it all in one night. The Messenger of Allah (blessings and peace of Allah be upon him) said: “I am afraid that you may live a long life and that you may get tired …” Classed as saheeh by al-Albaani in Saheeh Ibn Maajah.

He died in 63 AH; it was also said that he died in 65 or 68 AH, and there were other views.

Tahdheeb at-Tahdheeb (5/338)

17. ‘Uqbah ibn ‘Aamir (may Allah be pleased with him)

He learned the Quran from the Prophet (blessings and peace of Allah be upon him). He died in 58 AH.

Tahdheeb at-Tahdheeb (7/243)

18. ‘Aa’ishah bint Abi Bakr, the wife of the Prophet (blessings and peace of Allah be upon him and may Allah be pleased with her)

She learned the Quran from the Messenger of Allah (blessings and peace of Allah be upon him). She died in 57 AH.

Taqreeb at-Tahdheeb (p. 750)

19. Hafsah bint ‘Umar ibn al-Khattaab, the wife of the Prophet (blessings and peace of Allah be upon him and may Allah be pleased with her)

She learned the Quran from the Messenger of Allah (blessings and peace of Allah be upon him). She died in 45 AH.

Taqreeb at-Tahdheeb (p. 745)

20. Umm Salamah Hind bint Abi Umayyah, the wife of the Prophet (blessings and peace of Allah be upon him and may Allah be pleased with her)

She learned the Quran from the Messenger of Allah (blessings and peace of Allah be upon him). She died in Shawwaal 59 AH; it was also said that she died during the caliphate of Yazeed ibn Mu‘aawiyah, or in 62 AH.

Tahdheeb at-Tahdheeb (12/456)

See also the answer to question no. 211865

And Allah knows best.