Infinity is a paradox for itself. Mathematics says dimensions dont end. This use of dimension is the same use Thomas Banchoff and any other expert on higher dimensions would use. There is a final dimension, when you understand my theory of five dimensions, which I typed here for numerous times. Mathematics says dimensions don't end because mathematics is just a vast number of number combinations and there's no theory like in physics and philosophy.
Theory is the one that limits dimensions to five.
Firstly, there's five dimensions. I've got 3 different dictionaries here with me and there is no word trimension. And as you can see they have no connection to dimension. As you can see, there is nothing to support your speculations and dimensions of dimensions etc. As you saw in previous answer, there is only four dimensions known to the world.