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There aren't periodical payments involved with domain name purchase except yearly renewal. When you register your domain you can renew it for years to follow right then and there. I think in GoDaddy you can register it for something like 9 years if you so choose.
Once you have a domain name, you will need to have it hosted somewhere, which generally incurs additional costs, although here at Synthasite they will host your domain for free... pretty cool! Hope this helps a bit.