In a commercial scenario, take a look at yFiles for HTML:
It easily supports multi-graphs and does not look too ugly, I believe:
(These graphs have been laid out automatically, manual placement is also possible.)
Of course this is a matter of taste, so if you don't like the look, you can change any aspect of the visualization, like in the style tutorial.
The API offers full interactive editing capabilities and being a pure client-side solution, of course there is no API call count limit.
Large graphs are still a problem with todays' Javascript engines, but only if "large" means more than thousands of elements. With virtualization (considering only what is currently visible in the viewport during rendering), you can get good performance with thousands of elements.
Disclaimer: I work for the company that creates the library, on SO/SE, however I do not represent my employer. My posts and comments are my own.
