What was the origin of repeated confession of fault during the Act of Contrition? What was context and time of origin of the repeated chant of ".. through my fault, through my fault,through my most grievous fault" in the Act of Contrition said during the Holy Mass of Catholic Church (Latin rite) ?
The number 3 has many meanings in the Christian/Catholic tradition, and some orthodox faiths repeat many parts of the liturgy three times. It is often honoring the three persons of the Trinity, but here it is more likely harkening back to scripture, as the liturgy does often. It also worth noting that the Confeitor prayer is also simply part of Tradition (since about the 11th century in its modern form).

This blog post sums it up nicely:


  In the revised 2011 English translation of the Mass, the threefold
  refrain reappeared, and we are once again encouraged to symbolically
  strike our breasts each time we acknowledge our fault.
  
  Why was this restored? And why do we, at the beginning of the prayer,
  not only admit that we have sinned, but that we have greatly sinned?
  
  The first reason is that the Catholic Mass is thoroughly biblical. So
  let’s start by noting what a threefold repetition means in the Jewish
  mentality. In Hebrew, you can’t add endings to words to express
  superlatives or emphasis. There is no “big, bigger, biggest” in the
  language of the Old Testament. So how do you get the idea across that
  something is the best, worst, or extremely important? By repeating it
  three times. When Isaiah had a vision of the Lord attended by angels,
  they cried out, “Holy, Holy, Holy!” which we repeat at every Mass.
  Peter’s denied Jesus three times. The Lord later asked Peter, “Do you
  love me?” (John 21:15–17) not once, but three times.
  
  Therefore, a threefold admission of our fault is a biblical way to
  emphasize what is said at the start of the prayer—that we have
  “greatly” sinned. Sin is not just a casual affair, an inconsequential
  blunder like failing to dot an i or cross a t. Sin is a most serious
  matter. God has given us everything, even sacrificing his only Son for
  us. He deserves all our love, as we say in the act of contrition.
  
  We are commanded to love him with all of our heart, soul, mind, and
  strength. When we fail to do so by neglecting our duty or by doing
  something that hurts others, ourselves, and God’s honor, it is
  grievous. As we become aware of what we’ve done or failed to do, the
  proper response is contrition, the kind of sorrow that leads not to
  despair but to change.
  
  But why the striking of our breasts as we recall our fault? Is being
  Catholic all about beating ourselves up? For the answer, we again need
  to turn to the Scriptures. In Luke 18:9-14, we meet a very pious
  Pharisee who congratulates God on being so fortunate as to have such a
  worthy servant as himself. There is also a publican, a tax collector,
  who strikes his breast as he comes before the Lord saying, “Lord, be
  merciful to me, a sinner.”
  
  By striking our breast, we distance ourselves from the Pharisee and
  stand, or rather bow, with the publican, acknowledging our
  unworthiness before the awesome majesty and perfect holiness of the
  living God. It is not about self-hatred but about humility. And
  humility means getting in touch with reality. As we begin the liturgy,
  we pause to recall that we don’t deserve to be there. We are all
  publicans and prodigals whom a loving Father embraces and welcomes not
  because of our virtues but in spite of our sins.