In my experience, Jesus is generally understood to have used “seventy times seven” in poetic opposition to a small number: “seventy times seven” means “an impossibly large number”.
Matthew 18:21-22 is referenced at three places in the Catechism of the Catholic Church, and I've emphasised the parts relevant to this view. In each, it is emphasised that forgiveness should be limitless.
982. There is no offence, however serious, that the Church cannot forgive. "There is no one, however wicked and guilty, who may not confidently hope for forgiveness, provided his repentance is honest.527 Christ who died for all men desires that in his Church the gates of forgiveness should always be open to anyone who turns away from sin.528
2227. Children in turn contribute to the growth in holiness of their parents.36 Each and everyone should be generous and tireless in forgiving one another for offences, quarrels, injustices, and neglect. Mutual affection suggests this. the charity of Christ demands it.37
2845. There is no limit or measure to this essentially divine forgiveness,146 whether one speaks of "sins" as in Luke (11:4), "debts" as in Matthew (6:12). We are always debtors: "Owe no one anything, except to love one another."147 The communion of the Holy Trinity is the source and criterion of truth in every relationship. It is lived out in prayer, above all in the Eucharist.148
36 Gaudium et Spes 48 #4
37 Cf. Mt 18:21-22; Lk 17:4
146 Cf. Mt 18:21-22; Lk 17:3-4
147 Rom 13:8
148 Cf. Mt 5:23-24; 1 Jn 3:19-24
527 Roman Catechism I, 11, 5 (Catechism of Trent)
528 Cf. Mt 18:21-22