Web Hosting Service recommendations?- Email sending limits are a factor

I'm trying to set up a web-based business.  Short term, I just want a presence on the web.  Within a year, perhaps I'll be selling products --no not Viagra BTW, this is legit.  But right now I have a database of customers of several thousand.  I'm looking for a hosting service that doesn't say "You can only send 100 emails at a time." (UGGH!).  I like constant contact but my budget is SMALL.  Can't afford them.  My budget is VERY SMALL --I have a one-half of a shoestring budget.  Currently I'm not going to be collecting money on the web.  
Hubspot is a nice service for web-based businesses, but I'm not up to that point.
Recommendations please?  What is a semi-employed guy supposed to do?