This is a hard question to answer because social systems varied by time and place, and "peasant" wasn't even a word for most of the middle ages.

For example, in 14th century England there were unfree farmers tied to the land called villeins. It must have been very rare (I'm not sure it ever happened) for a villein to become a noble. A step higher, yeomen were free farmers who owned their land--no downtrodden unwashed masses. A yeoman who acquired enough property or performed a major service would be made into a knight, but that was non-hereditary, so it's not the same as an earl.

There were about 100 nobles, and the lowest rank, the baron, had an income of at least 300 pounds per year. There were about 1000 knights, having an income of at least 40 pounds. So to go from below-knight to baron you'd have to increase your wealth by 10x without modern economies of scale. And a 39-pound-per-year landowner would be considered an esquire or gentleman, who you probably don't consider a peasant. A regular yeoman would start much lower. (Source: The Time Traveler's Guide to Medieval England. Great book, btw.)