Basic HTML view lets you access your Gmail messages from almost any computer running almost any web browser, even old ones (not just IE5.5+, Mozilla, and Safari). Especially great for traveling, since you never know what kind of browser that internet cafe in Siberia is going to have.
Standard accessibility good practice is to avoid text-only versions of pages like the plague, but in my opinion the distinction between web content and a web application make simplified versions of Ajax style applications an acceptable compromise.
More on AJAX and accessibility can be found over on Standards-Schmandards.com.
The usability side of Ajax is interesting as well, mainly because Ajax tricks can severely alter the expected behaviour of the browser. Thomas Baekdal put out a set of suggested XMLHttpRequest Usability Guidelines a few weeks ago, and recently followed it up with an article putting his ideas to practise in the context of a neat business card designer example.
Finally, for those who really can’t get enough Ajax Mike Papageorge has assembled a comprehensive list of working examples.