Will a pony be a full grown horse?

No. A pony is not a horse at all. It is a member of the Equine family but it is not a horse. Here are your classifications, you have a miniature horse which is any horse under 48 inches of height, then you have a pony which stands at a height of anywhere between 14.2 hands and 48 inches, then you have your regular horse that is anything about 14.2 hands.So I have a little different take on this thread. A pony is any animal (horse) that

is under 14.3. Anything above 14.3 a horse. Any breed can produce a pony sized animal.

Miniature horses are called horses because they were "bred down" to the size they are without the use of any of the pony breeds...or so the story goes.

Miniature horse people like the idea of miniatures as a small horse as opposed to a pony hence the name. However, from a technical standpoint, based only on height, they are a pony.