Those days are over, thanks to two totally free services.
The first I'll address is Photobucket. This is a free donor-supported service that gives users up to 100 MB of storage space and 2,500 MB of bandwidth. That's pretty good for a free service, but not a huge amount, so users have to be a little careful.
It's great for low-resolution images that are not associated with one particular post, such as buttons and banners.
I would not use it for anything that is ripe for hotlinking by other people. For example, if you upload a picture that other bloggers might be tempted to swipe by copying the image tag, then your bandwidth might get used up and then all your imaged will be lost and won't appear on your blog.
The other free service is called Hello! It's available from Google, the same people who run Blogger. It allows people who post through Blogger to upload one single image in the form of a Blogger post.
There are limitations. First is that the images must be used in a post and there only one post. Users must download a small program from their Website, and there are some that consider the program to be spyware. The interface allows for words and HTML to be uploaded either above or below the image. The interface is very clunky, in my humble opinion, and posts require some tinkering and fixing later by logging in directly to the Blogger site.
The upside is that I cannot find that there are any limits on image storage or on bandwidth, and the images are stored on Blogger's services. Also, Photobucket doesn't allow adult content while it appears that Blogger has any objection. So if that is your thing ...