The chromosomes actually double BEFORE meiosis - during S-phase (end of interphase).
You don't get too many chromosomes because meiosis has two parts - in part 1 you have the doubling of the chromosomes before (during S-phase) but as soon as the 2 daughter cells are formed at the end of meiosis 1 (telophase 1), the cells go straight into another division cycle (meiosis 2) WITHOUT duplicating the chromosomes...so you end up with HALF of the number of chromosomes you started with.