In the beginning of meiosis, there is one cell. During meiosis, 1 cell divides into 4 cells. Does each of these 4 cells divide each into 4 more cells?
No, meiosis is not a cycle like mitosis. In mitosis, haploid or diploid cells divide to create two genetically identical cells, and this process can go on and on. On the other hand, meiosis results in 4 genetically unique daughter cells which are also haploid. If a haploid cell was to try to undergo meiosis, there would not be enough genetic material and you would end up with half-chromatids.
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In species where an individual can be haploid (for example in haplodiploidy) then cells produced by meiosis do multiply by mitosis in order for the individual to grow.
From a brief scan the Wikipedia article on ploidy nicely summarises the various possibilities.