The people who ask this are rarely asking about one missed Asr. They are asking about a period — a few years, sometimes a decade — and the arithmetic frightens them into doing nothing, which is the one outcome every scholar on every side of this would call the worst.
The part nobody disputes
Where a prayer is missed through sleep or forgetfulness, it is prayed when the person wakes or remembers, and there is no sin in it. The hadith is explicit: whoever sleeps through a prayer or forgets it should pray it when he remembers it, and there is no expiation for it beyond that.
The same applies to a prayer missed through unconsciousness, and to a woman's prayers during menstruation, which are not made up at all — the obligation lapses rather than accumulating. Fasts missed for that reason are made up; prayers are not. It is a distinction people frequently get backwards.
Where the disagreement is
The hard case is prayer abandoned deliberately, over a long period, by someone who knew and could have prayed.
The four schools hold that these are made up. The reasoning runs from the lesser to the greater: if someone excused by sleep must still pray it, someone with no excuse can hardly be under a lighter obligation. This is the position of the overwhelming majority, it is what a local imam will almost certainly tell you, and it is the safer course in the sense that acting on it cannot leave you having done too little.
The minority position — held by Ibn Hazm, and by Ibn Taymiyyah and Ibn al-Qayyim after him — is that a prayer deliberately abandoned outside its time cannot be made up, because the time was part of the act and it is gone. On this view, making them up is not merely unnecessary but not possible, and what is required instead is sincere repentance, a firm resolve never to abandon another, and an increase in voluntary prayer to make up the deficiency.
Both positions read the same hadith. The majority reads its naming of sleep and forgetfulness as an example. The minority reads the naming as a limit — these two cases, and by implication not others.
What they agree on, which is most of it
Neither position offers comfort about the years. Neither treats it as a small matter. And both arrive at the same three instructions: repent sincerely, never abandon another prayer from today, and pray more than the minimum.
Anyone using the minority view as a reason to do nothing has misunderstood it, since it asks for more voluntary prayer, not less prayer. And anyone using the majority view to conclude that the debt is unpayable and there is no point beginning has drawn the opposite of the conclusion it supports.
If you are making them up, how
The practical advice is consistent and it is not what people expect.
- Estimate honestly rather than exactly. Nobody can reconstruct which prayers were missed in 2014. A considered estimate made in good faith is what the scholars ask for, and the paralysis of trying to count precisely is itself a reason people never begin.
- Attach them to the prayers you are already praying. One or two make-up prayers after each of the five is a rate that finishes a large number over a few years without becoming an ordeal that gets abandoned in a fortnight.
- On order: the Hanafi position requires the missed prayers to be made up in sequence, but that requirement lapses once the number exceeds a day and night — six prayers — which is why it does not apply to the case being asked about here. The Shafi'i position treats order as recommended rather than required.
- The current prayer comes first when its time is running out. A make-up prayer never displaces a prayer in its own time.
- There is no expiation by feeding or by money in place of a missed prayer. That mechanism exists for missed fasts, not for prayers, and being told otherwise is the most common misinformation on this subject.
The advice that matters most
Begin. On the majority position you are discharging a real obligation, and on the minority position you are increasing voluntary prayer, which it explicitly asks for. There is no reading of the disagreement on which starting is wrong, and there is no reading on which continuing to miss them is right.
If the number is large enough to feel hopeless, take it to a scholar who knows your circumstances. That is a better use of the question than a search result — this one included.
This is not a fatwa. It sets out the positions scholars hold and where they part, so that you know what you are asking about. Your own case turns on facts a page cannot see — put it to someone qualified.