Must the amplitude of a driving force increase so as to match the increase in the amplitude of the standing wave it generates? If not, how can standing waves of increasing amplitude form if the amplitude of the waves generated by the source does not match the amplitude of the waves prevailing in the system?
Your 2nd sentence and comment suggest that you are thinking about a standing wave on a string, with the driving force being provided by a mechanical oscillator at one end, like a hand shaking a rope. You are perhaps confusing the amplitude of the oscillator (measured in metres) with the amplitude of the force (measured in Newtons).
In this case there is an unnecessary complication. The oscillator is performing the multiple functions of (i) keeping the string taut, (ii) providing a restraint from which the wave is reflected, while also (iii) doing work on the string so as to input energy. As the amplitude of the waves increase, and the accelerations of points on the string increase, then the reaction forces on the restraints at both ends of the string will also increase - but not in proportion to the amplitude.
We need to consider a simpler case, in which the oscillator is not doing several jobs at the same time, only providing a periodic push.
The driving force could be provided without any mechanical contact. For example, parallel electrified plates above and below the string providing a force on a small, insulated, charged section of the string. Then the force on the charge does not depend on the position, velocity or acceleration of the charge, only on the uniform electric field between the plates. This field with constant amplitude would only need to oscillate at the natural frequency of the string in order to create or increase the amplitude of standing waves.
As with all harmonic oscillators which are driven at resonance (ie at the natural frequency of the system), the amplitude of oscillations continues to increase although the amplitude of the driving force is constant, until damping removes energy from the system as fast as the driving force is putting it in.
See Standing waves due to two counter-propagating travelling waves of different amplitude and also How are standing waves a result of constructive and destructive interferences? which includes a demo in Brionius' answer.