I've been married twice non-sacramentally, divorced both times, and one of the marriages was annulled. May I return to the Church? I don't intend to remarry.
Actually, from what you say, you never left the Church in the first place. Many people think that by entering into an invalid marriage or by being divorced they have left the Church. This is rarely the case. One leaves the Church by deliberately denying the Catholic Faith in a stable and public way, or by being excommunicated for some misdeed for which separation from the Church is the medicinal, spiritual penalty. But in Church law, excommunication is not incurred by an invalid marriage or by divorce and remarriage. Committing a sin is not the same as leaving the Church.
Let's assume the invalid unions you entered into were the result of your own sin. You would still be a member of the Church, and all you would need to do is to go to confession with true repentance, and you could return to the sacraments and live a full, Catholic life. So by all means, consider yourself a member of the Church, for that is what you are, and get to confession so as to be ready to receive the graces of the Holy Eucharist.
If in the future you do want to marry, or even to marry validly one of your previous partners, the previous unions would not stand in the way, since they were not the sacrament of matrimony. You would need to talk to your parish priest to arrange this.