Do there exist algorithms for which the Big-O cannot be found out?
There are some algorithms for which, as of today, we are not able to prove termination, much less to bound complexity.

A simple case is given by the Collatz function: in pseudocode

function Collatz(n):
  if n &lt;= 1  then return 1
  if even n then return 1 + Collatz(n/2)
  return 1 + Collatz(3*n + 1)


It is believed that, for any natural $n$, Collatz(n) will eventually stop recursing. However, we do not know whether this is is the case. Consequently, we also do not know an upper bound for the time complexity of this algorithm.