Non constructible functions and anomalous results
Borodin's Gap Theorem: For every total computable function $g(n) \geq n$, there is a total computable function $t$ such that $DTIME[g(t(n))] = DTIME[t(n)]$. 

In fact, this holds for any Blum complexity measure in place of $DTIME$. 

See also the wikipedia page and references therein.