Would anyone recommend a particular JavaScript charting library - specifically one that doesn't use flash at all?
There is a growing number of Open Source and commercial solutions for pure JavaScript charting that do not require Flash. In this response I will only present Open Source options.
There are 2 main classes of JavaScript solutions for graphics that do not require Flash:
There are pros and cons of both approaches but for a charting library I would recommend the later because it is well integrated with DOM, allowing to manipulate charts elements with the DOM, and most importantly setting DOM events. By contrast Canvas charting libraries must reinvent the DOM wheel to manage events. So unless you intend to build static graphs with no event handling, SVG/VML solutions should be better.
For SVG/VML solutions there are many options, including:
Raphael is a very active, well maintained, and mature, open-source graphic library with very good cross-browser support including IE 6 to 8, Firefox, Opera, Safari, Chrome, and Konqueror. Raphael does not depend on any JavaScript framework and therefore can be used with Prototype, jQuery, Dojo, Mootools, etc...
There are a number of charting libraries based on Raphael, including (but not limited to):
Disclosure: I am the developer of one of the Ico forks on github.