What is the maximum distance for class C service?
There is no such thing as "Class C radar service".  "Class C airspace" and "radar service" are mostly orthogonal.  "Flight Following" is just shorthand for getting radar service when you're not required to.

Terminal areas around a Class C airport generally comprise a 5-mile "inner ring", 10-mile "outer ring" and 20-mile "outer area".  Approach is guaranteed to control (at least) that much airspace, regardless of class.  Within the airspace they control, they provide (roughly) the same radar services regardless of whether you're in Class C or Class E at any given moment.  The only practical difference is that in Class E, you're not required to talk to them, whereas in Class C, you are.

The reason the box says "CTC KNOXVILLE APP WITHIN 20NM" is that 20 miles guarantees you're within the "outer area" and thus Approach's airspace.  If you contacted them at 30 miles out, you may be Center's airspace, in which case Approach would need to redirect you to Center, and then Center would hand you back to Approach a few minutes later.  That's a waste of both your time and the controllers' time, so the chart reminds you of the minimum range to help you avoid that.

There is no theoretical maximum, and many Approach facilities do in fact control more (sometimes far more) airspace than the minimum.  However, the FAA does not publish the actual sector boundaries, so the best practice is to assume no more than the minimums&mdash;or get Flight Following for your entire trip, in which case you don't have to worry about it at all.