Is every algorithm on bounded resources O(1)
By definition, every algorithm (that always halts) on a finite domain has $O(1)$ running-time: pick $n_0 &gt; \max \{ |x| \mid x \in D\}$ with $D$ the input domain of the algorithm. 

However, that fact is utterly uninteresting. Asymptotics are designed to help make statements for "large n", expressed through a limit process; they are simply meaningless if the domain is bounded, or the inputs you see in practise are "small".

As a consequence, if you are concerned with finite domains you'll have to use different mathematical notions, and potentially different machine models.