The simple answer to your question is 'No'. You can host your site anywhere you like.
However, you should look into things such as address masking. For example, if you host your site with 'mythicalisp.com', it's 'real' address will be something like www.mythicalisp.com/users/a/amonty. If you simply purchase the domain name www.amonty.com, your traffic will automatically be routed to the 'real' address but, unless you pay for address masking, the 'real' address will then appear in the visitor's browser address bar. If you want 'www.amonty.com' to appear, you require address masking.
You might also need to consider whether you want to pay for an e-mail service from the company which sells you the domain name. The simplest option is 'mail forwarding'. This allows you to have mail addressed to, say,
[email protected] forwarded to your personal e-mail account. The problem with this is that your replies will show your personal address, rather than the amonty.com one. It would be much better to pay for a full mail server service. Then your business mail could be kept completely separate from your personal mail.
Chris