It really depends on how busy of a site you have...But, a reseller account will probably do, although it depends on each host.
There is an option in "between" reseller accounts and dedicated servers called "virtual servers", where you typically have 1/5 - 1/4 of a server and they are manages for you.
Virtual servers are typically about $50 a month, and are a good balance between power and control and price.
Another option is to us the free program HTtrack and "suck" the dynamic pages to your hard drive, making them static html pages, then uploading to various hosts. This will reduce the server resources needed.