No, WordPress don't allow html on blogs that are hosted on WordPress.com. I think the rest do to some extent. If you want to go the way of running the blog entirely yourself, there is webhosting that allows blog software installation - usually with something called 'Fantastico'.
'richie' in an early post in this thread says he runs the hosting company - Mydosty. I haven't used it and this isn't a personal recommendation but the service looks cost effective for what is on offer. It has Fantastico and judging by the control panel it allows the installation of WordPress and three other blog softwares. Although you can also install your own, yourself
Try and find a webhost that has a monthly bandwidth allowance of at least 3GB. Some offer a starter of 500MB, which is ok until someone mentions you and links to your blog bringing too much traffic. So 3-5GB a month is a good minimum.