How to understand the open-loop gain of a op-amp?
•How is this open-loop gain measured? Do we use some kind of special
  voltage meter to measure the voltage across the output and the ground?
  But if so, it is not open-loop at all. It's kind of a dilemma. Or do I
  derail from the correct path of Electrical Engineering thinking?


Measurement of open-loop gain is (in principle) straightforward. You apply a very small input voltage and measure the output Generally, this is best done using a low-frequency AC signal, to allow distinguishing gain from offset. And while it's straightforward, that's not the same as easy. Shielding against noise pickup, particularly 50/60 Hz power line radiation, can be a challenge.


  •What's the point of the open-loop gain if the actual output voltage
  can never be that high?


The answer is simply that the feedback network acts to keep the voltage difference AT THE OP AMP INPUT very small, typically in the microvolt to millivolt range. If you look at a simple inverting op amp circuit, for instance, 







simulate this circuit &ndash; Schematic created using CircuitLab

if the op amp has an open loop gain of 1 million, since the output is -10 volts, the voltage difference at the input will be about 10 uV. In fact, it is the very small input voltage which allows an approximate gain formula $$G = -\frac{R2}{R1}$$ to be usefully correct, since the - input can be considered to be held at ground. Assuming a high input impedance gives identical currents in the resistors, and Ohm's Law then provides the voltages across the resistors.

But the heart of the matter is that a properly-working op amp will not, in fact, see more than a few millivolts across its inputs, or less if the open loop gain is very large. In your example of 1 volt into an op amp, the output will be stuck somewhere near one of the power supply voltages, and the term for this is "saturated output". This is actually quite useful when making comparators, but not so much for linear amplifiers.