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I have a simple Newtonian reflector telescope. Using it, I am able to see the Galilean moons of Jupiter. However, Jupiter has much more moons than that (Wikipedia says 67 have been discovered this far). What kind of telescope I would need to see most or all of them?

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Unless you're really rich, unfortunately you won't be able to see all of them.

Jupiter's fifth largest Moon, Amalthea, has an apparent magnitude of $m$ = 14.1. Comparing this to the magnitude of Europa, the dimmest of the Galilean moons, which is 5.3, tells us that Amalthea is roughly 3000 times less bright. Your telescope thus needs to have an area 3000 (or radius ~55) times larger for Amalthea to have the same apparent brightness.

In general, without a camera on your telescope, the dimmest object you can see depends on your vision, but on average, humans are able to see objects of magnitude 6. That means that in order to detect an object of magnitude $m_\mathrm{obj}$, you need a light-collecting area which is larger than you pupil by a factor of $$f = 10^{(m_\mathrm{obj}-6)/2.5}.$$ Thus, in order to be able to just barely detect Amalthea, you need a telescope which is larger than you pupil (radius = 6 mm) by a factor of ~1738, i.e. has a radius of 25 cm.

It quickly becomes really difficult to see them; for instance, Elara, which is the eighth largest moon, has an apparent magnitude of 16.3 and thus requires a 70 cm telescope. And remember, this is just on the threshold of what you can see.

Of course, if you mount a camera to your telescope and take images with long exposure times, you can get away with smaller telescopes. In fact all moons smaller than Amalthea were discovered this way, some of them by cameras on the Voyager space probes.

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