There's a lot of reviews of low-cost private server providers
Just a note that lowendbox only shows VPS offers at $7 and less. Sounds like what you're looking for.
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Depends on your standards and needs.... personally I have bad experiences with sub 10usd VPS's, but as a mail server or small website they might be sufficient. But there might be some perfomrance vps's to be found even in this price range if for example their node isn't loaded with other clients yet and whatnot.