Example of language accepted by Turing Machine but not by Linear Bounded Automata ? Is there any EXPSPACE language?
Consider the language of pairs $(n,T)$ where $n$ is an integer and $T$ is a description of a Turing machine that, when started on an empty tape, eventually moves farther than $2^n$ squares away from the origin.
The empty language is in EXPSPACE. Wikipedia's EXPSPACE article mentions a known EXPSPACE-complete problem.
See also the space hierarchy theorem which gives a simple diagonal construction of languages tailored to have specific space complexities.