If amplitude (say voltage) of a 60Hz sine wave increases, doesn't rise time increase?
Risetime and slope are two different things.  A sinewave of higher amplitude will have a steeper slope, but the rise time will be the same as it takes the same amount of time to get from the lowest point to the highest point.  Also, a sinewave by definition has only one frequency component, regardless of its amplitude.  A sinewave is fully defined by a frequency, amplitude, and phase offset.  

Waves which are not sinewaves can be thought of as a summation of a bunch of sinewaves.  For instance, a square wave contains all of the odd harmonics - 3, 5, 7, 9, etc. times the fundamental frequency.  The more of these harmonics are present, the flatter the tops and bottoms of the wave become, and the steeper the transitions become.  With a square wave, the sharper the edge, the more harmonics are required to represent it.  However, this has nothing to do with the amplitude - if you double the amplitude, you don't double the number of frequencies required, you just double the amplitude of all of the harmonics.  

However: The slope of a sinewave does change as you increase the amplitude, and slope can sometimes be an issue!  Op amps generally have a specified output slew rate that limits the slope of the signal that it can reproduce.  It turns out that if you build an op amp circuit that has enough gain, you can run into the slew rate limit before the 3 dB bandwidth limit with a large enough signal.  When this happens, your nice sinewave will become a triangle wave at the output.  Naturally, this distortion will create more frequency components than the original signal.