Define $T\colon \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ by:
$T(k,n) = 2^{T(k-1,n)}$
$T(0,n) = 1$
$T(1,n) = 2^n$
$T(2,n) = 2^{2^n}$
And denote the W.C. (worst case) time of computing $f(n)$ as $WC(f(n))$
Can we find computable functions $f_i\colon \mathbb{N} \rightarrow \mathbb{N}$ such that computing $f_i(n)$ takes $\Omega(T(i,n))$ W.C. and such that for every other computable function $f \equiv f_i$ (I mean $f(n) = f_i(n) \forall n \in \mathbb{N}$), we have $\lim_{n \rightarrow \infty} \frac{WC(f_i(n))}{WC(f(n))} \neq 0$?
I am in a sense trying to find a concrete example of functions that takes at least X time to compute, where X can be as large (asymptotically) as we want.
I was thinking maybe $f_i(n) =$ the $T(i,n)$'th digit of $\pi$, but not quite sure this fulfills the second condition.