What exactly do they mean when you say you got blocked using google because they have limits? MailChimp have limits? Google has limits? Could you be a bit more specific?
I use MailChimp which is free, and yes, there is a limit of 2000 subscribers and a send limit of something like 12,000 emails. It's a great piece of software and great for anyone on a budget. Tell me what the issue is exactly and I'll bet I can help.