So it turns out you can add a Flickr photos badge to any Typepad blog without having to edit any templates. Here’s a quick, step by step guide of how to do it:

Creating the Flickr Badge
1. Login to flickr
2. Click on Yours in the upper left nav, then Flickr Badge on the lower right.
3. Configure your badge.
4. Copy the Javascript in your clipboard, save it somewhere if you’re worried about it.

Plugging it into Typepad
1. Login to Typepad
2. Click on TypeLists
3. Create a new Link list in the right hand column
4. Click on Add a New Item
5. Paste the Flickr Javascript code into the Link title field
6. Close the Add Link window.
7. Click on Add this link to your weblog(s) or About Page
8. Select your weblog. Save changes.
9. You’re done. Yay! Check your blog.

Update
If you don’t like the extra anchor tag that is produced, you can paste the JS code into the Link Description field. Click on the Configure tab for your list, and select Display Notes as text. Thanks Anil!