At what phase of meiosis is the number of chromosomes per cell doubled? (Think carefully about this.) What happens after that so that the number of chromosomes per cell does not end up being higher?

I'm going to attempt to answer this; as far as I know, meiosis itself never involves chromosome replication.

Rather, replication occurs during the S-phase of interphase.

The cell then divides via meiosis to prevent further replication.