You could argue for more exotic rock types but that would imply a slightly more complex geological history (which is totally possible). For example, you could have uplifted a huge package of sedimentary rocks without any volcanism involved. Or you could have simply eroded everything around leaving large rock packages like the Venezuelan Tepuis (below).

In general, the rocks deep inside a mountain (or simply super old rocks) have also undergone a lot of compression, so sedimentary rocks will be unlikely as you would have converted them into metamorphic rocks, e.g. limestones would be marble, sandstones would be quartzites (like the Tepuis), shales would be slates (or schists, etc.), etc.

Similarly, metamorphic rocks that have undergone a higher degree of compressional and thermal stress (i.e. deeper in the mountain) will grow larger crystals.

Of course, like anything in geology, other complex scenarios do exist where these rules of thumb don't work. But in general, this should get you started.

Anyway, hope that helps.