Many op amp circuits are designed so that they would yield a known finite gain if constructed using ideal components including an infinite-gain op amp. In practice, such circuits will always be constructed with non-ideal components, and their behavior will not quite match what would have resulted from ideal components. Consider a very basic amplifier:
simulate this circuit – Schematic created using CircuitLab
When using ideal components, the gain will be (R1+R2)/R2; I'll call that the "nominal gain". In an actual circuit, if an op amp has a constant open-loop gain, the gain will be 1/(R2/(R1+R2) + 1/opAmpGain). If the open-loop gain of the op amp is much larger than (R1+R2)/R2, then 1/opAmpGain is going to be very small relative to R2/(R1+R2), and its exact value won't matter much. Further, even if the open-loop gain might vary due to factors like frequency or--even worse--the input voltage, the maximum and minimum gain for the circuit would be relatively close. For example, if the open-loop gain might vary between 500x and 1000000X, the net gain of the circuit would range from about 9.8x to 10x. More variation than might be ideal for some uses, but still pretty small.
If R1 were changed to 99K (changing the nominal gain from 10x to 100x), then the circuit's sensitivity to the actual gain of the op amp would be increased more than tenfold. The same variation in the op amp's actual gain would cause the circuit's net gain to range from about 83x to 100x--a much larger variation. If one were to instead cascade the circuit shown below (for 10x gain) with a second copy, the resulting circuit would have a gain that could range from about 96x to 100x. A larger degree of relative uncertainty than when using one copy of that circuit, but much smaller than when trying to achieve 100x gain in one stage.
A gain of 60dB would entail a 1000:1 voltage gain. While one op amp with a high enough open-loop gain to make a 1000:1 nominal gain practical at audio frequencies might be cheaper than two op amps with slightly-inferior specifications, op amps that will work well at such higher gains are apt to be much more expensive. At some level of gain, using two cheaper amps will be more practical than using one amp which is of sufficient quality to work well at the higher gain.