Can the runtime of functions with no loops change with the number of calls?
Depends on what something and do this are. In particular, do we have to assume global (or object) state?

Consider this:

int switch(int param) {
  if ( !set.contains(param) )
    set.add(param);
  else
    set.remove(param);
}


Now the runtime clearly depends on the implementation of set and the number of entries it has when switch is called. Calling switch $n$ times starting with an empty set and using pairwise distinct parameters, you might get $\omega(n)$ runtime in total, hence not all calls run in time $O(1)$.

On the other hand, if somefunction does not access state outside of its scope -- assuming that the function itself does not have persistent state -- the runtime can only depend on param. That does not mean runtime is in $O(1)$ -- something and do this may still do non-constant things. Call methods, recurse, etc.

Note that depending on the cost model you use, even adding two numbers has non-constant cost.