Sup

This is a whole other question, because you're now taking a mathematical aspect and assuming it makes sense to apply it in a physical situation as a "description" of the system. As for your question, the infinite derivative of the position x(t) does not make sense and is not used in modern physics - and no, as the infinite derivative is not defined, we cannot do so.

do not like.

You can keep taking derivatives forever, and when you finally do reach the infinith derivative, it can be whatever you want. Anything continuous on C^(inf) has such a derivative, but since it's continuous the derivative is not infinite. And anything past C^(3) isn't really useful, but special functions that can be described by infinite power series (trig, trig h, bessel.... most others) will have a infinith derivative.