You should handle exceptions in your callbacks. jQuery will halt whenever an unhandled exception bubbles up which makes it much easier to spot bugs in applications.

You could also use a prefilter rather than ajaxStart and ajaxStop:

$.ajaxPrefilter(function( options, _, jqXHR ) {
    startAnimation();
    jqXHR.always( stopAnimation );
});

We always stop animation, but why is it that the global ajax start/stop no longer get fired after an exception? Shouldn't they continue to work afterwards on subsequent calls?

It's because we have a global counter that is not properly reset to 0 because the exception bubbled. One of the reasons why I always tell people to prefer prefilters whenever applicable. ajaxStart and ajaxStop take concurrent requests into account, a feature that is actually rarely useful and can easily be handled by a prefilter using a simple counter.

You could try and reset jQuery.active to 0 but I'm not sure that's a road you want to get down to.