I recall seeing in the Rema somewhere that he says the minhag is to wash one’s hands 3 times for bread.
Shalom, Thank you for your question. The Shulchan Aruch (Orech Haim, 162, 2) writes “One who washes their hands [should first] pour on them a small amount of water, less than a revi'it, in order to remove any dirt or other substances [from their hands]. Aftherwards, one pours water a second time [on the hands], and this water is also impure. The after this one pours on them a third time in order to purify the water that was on the hands. But if one's hands were clean, and had no intervening substances on them, one pours over both hands a revi'it of water, and one does not need a second pouring of water”. To this the Rema adds, “And so too, if one has much water, they can wash [their hands] a little at the start to clean them, and afterwards pour a revi'itThe in one time, and one would not need a third washing”. The Mishna Brurah (ibid 21) quotes many opinions who hold that even if one has clean hands, and pours a revi'it of water over them, they should none the less pour another time over their hands, as one always needs two pourings (on each hand). He advises one to follow this opinion, unless one is short of water etc. So, in summary, the Shulchan Aruch and Rema both hold that after one has clean hands, one only needs to pour one time on each hand, being careful to pour a revi'it of water on each hand (- there is a way to pour on both hands together, but we won't go into that now). Their are other authorities who hold that one should always pour twice on each hand, and the Mishna Brurah rules like this. Their are those (see Kaf HaHaim ibid 14) who quote kabbalistic source that hold one should pour three times over each hand. I did not see anywhere in the Rema where he writes that one whould pour three times on each hand – only in the Shulchan Aruch when one's hands are dirty (as we quoted above). I hope that this, and the original response are of use to you. Blessings.