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I apologize for my lack of basic understanding of gyro-physics, I tried looking up internet, but couldn't find any answer for this particular question.

I have been told that if I apply torque to the axis of a gyro whose axis is horizontal, then the gyro's axis will precess horizontally and will remain parallel to the ground.

But what happens when I increase the torque to very high value? would the gyro precess faster or turn vertical?

My common sense is telling me that it will turn vertical at high values of torque, but I couldn't find any reference to it online.

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I can confirm your common sense.

The general rule that you refer to is an approximation, valid only for a torque that is small enough. So what is small enough here? Given a particular spinning rate there is for each rate of precession a corresponding torque. A torque that is such that the corresponding precession rate is, say, 1 percent of the spin rate is probably small enough.

As you point out, a sufficiently large torque will simply topple a spinning gyroscope.

The following two resources provide all the infromation that is needed to understand gyroscopic precession.
A 2012 answer here on physics.stackexchange, by me, to the question: What determines the direction of precession of a gyroscope?
Paper by Svilen Kostov and Daniel Hammer: 'It has to go down in order to go around'.
Kostov and Hammer discuss how a spinning gyroscope responds to a torque, and they have performed a benchtop experiment to verify the expected behavior

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