They don’t stay tiny for long. My sons adored their fluffy soft blankets.
When they were about a year old, each in turn, I bought a twin-size cotton thermal blanket in a nice color, cut it in half, and stitched satin blanket binding along the raw edge. This gave us two identical “blankies.” They became very attached to their special blankets, snuggled with them, carried them around, used them in various kinds of play, and so on.
It was nice that they were big, lightweight, and soft, but the best thing was that there was always one available and I could wash the other without triggering a crisis. And if anything ever did happen to one, we had a backup. (Let’s hear it for redundant systems!)
My 27-year-old son, now a lawyer, still has the tattered remains of one of his pair of blankets rolled up in his bureau at our house even though he no longer lives with us. I think the other one fell apart after he tried to make a tent out of it when he was in middle school.