He fasted before he reached puberty and he forgot to make up some days. Can he make them up after reaching puberty?

Firstly: 

A child is not obliged to do the obligatory duties of Islam until after he reaches puberty, because the Prophet (peace and blessings of Allaah be upon him) said: “The Pen has been lifted from three: from the insane person who has no control over his reason until he recovers, from the sleeper until he wakes up, and from the child until he reaches puberty.” Narrated by Abu Dawood (4399); classed as saheeh by al-Albaani in Saheeh Abi Dawood. Whatever the child does of obligatory duties or mustahabb deeds before reaching puberty, he will be rewarded for them. Whoever performs Hajj when he is a child or fasts or prays, the reward for that will be written for him in full, but his Hajj does not count as the obligatory Hajj or Islam, and he has to do Hajj again after he grows up. 

It was narrated from Ibn ‘Abbaas (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) met some riders in al-Rawha’ and he said: Who are these people?” They said: We are Muslims. They said: Who are you? He said: “The Messenger of Allaah.” A woman lifted up a child and said: Is Hajj valid for this one? He said: “Yes, and you will have the reward.” Narrated by Muslim (1336). 

Al-Nawawi (may Allaah have mercy on him) said: 

In this report is the evidence for al-Shaafa’i, Maalik, Ahmad and the majority of scholars, who said that the Hajj of a child counts and is valid, and he will be rewarded for it, but it does not count as the obligatory Hajj of Islam, rather it counts as a voluntary Hajj. This hadeeth is clear on this issue. End quote. 

Sharh al-Nawawi (9/99). 

Al-Khattaabi said: 

It counts as Hajj for him with regard to attaining the reward, but it is not to be counted as his obligatory Hajj if he lives until the age of puberty and reaches manhood. This is like prayer, which he (the minor) should be told to do if he is able to, although it is not obligatory for him, but the reward will still be written for him, by the bounty of Allaah, and it will also be written for the one who tells him to pray and teaches him to pray. 

See: ‘Awn al-Ma’bood (5/110). 

He is not enjoined to make up the fasts that he missed when he was a child, and he is not to be enjoined to make up any prayers that he missed. 

Ibn Qudaamah (may Allaah have mercy on him) said: 

As for the days that he missed (of Ramadaan) before reaching puberty, he does not have to make them up, whether he fasted them or not. This is the view of the majority of scholars. Al-Mughni (3/94). 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) said: 

It is not obligatory for a child to fast, but his guardian should tell him to fast so that he will get used to it. Fasting – for the child who has not yet reached puberty – is Sunnah, and he will be rewarded for fasting, but there is no sin on him if he does not fast. End quote. 

Fiqh al-‘Ibaadaat (p. 186). 

Based on this, you do not have to make up these three days that you did not fast, because fasting was not obligatory for you. But if you want to make them up there is nothing wrong with that. You should only make up the fasts, one day for each day missed, and you do not have to do anything else. 

Secondly: 

You say in your question that you are now 16 years old and have not reached puberty. 

But you have in fact reached puberty now, because puberty is reached when one of three things happens which in the case of the male are: 

1-Emission of maniy (semen)

2-Growth of coarse hair around the private part (pubic hair)

3-Reaching the age of fifteen.

In the case of the female, there is a fourth sign which is the onset of menstruation. 

When one of the signs is established, then puberty has been reached. It is not essential for all of these signs to be present. 

See also question no. 70425. 

As you have reached the age of sixteen, you are considered to be an adult, so you should act on that basis. The time of childhood and not being accountable has come to an end, and the angels record everything that the adult does, good or bad. 

“So whosoever does good equal to the weight of an atom (or a small ant) shall see it.

8. And whosoever does evil equal to the weight of an atom (or a small ant) shall see it”

[al-Zalzalah 99:7-8]

And Allaah knows best.