Examples of languages not decidable by a TM using certain upper bounds on space/time
Look up the time hierarchy and space hierarchy theorems. They give examples of languages which can be computed using $f(n)$ time (or space), and require more or less $f(n)$ time (or space). These languages are basically the following:


  Descriptions of Turing machines that halt after time $f(n)$ (after using space $f(n)$).


Perhaps you won't consider these languages natural. Unfortunately, the best lower bounds known for natural languages are currently only $\Omega(n)$ (for time) and probably even worse for space. However, it is conjectured that NP-hard problems require exponential time (this is known as the Exponential Time Hypothesis).