What is an O(1) operation performed infinite times? An assigning operation or an if condition is considered O(1) time complexity. Let's say I hypothetically have infinite if conditions or assignments. What would the total time complexity be?
If, in the worst case, an algorithm executes an infinite number of steps on an input of finite size, the algorithm's runtime is not bound from above by any function and so can be neither big-Oh nor little-Oh of any function. It can still be big-Omega or little-Omega of some function(s).
An algorithm whose worst-case behavior cannot be bound from above by any function could still, hypothetically, have best-case or even average-case (for specific certain distributions of input!) which is bound from above by some function.