There are examples of algorithm implementations that contain nested loops but are of complexity O(n), and some of them have corresponding implementations that contain no nested loops. So here comes a question, can all such implementations be simplified or converted to an implementation with only top layer loops? Namely, can all problems that have an $O(n)$ algorithm be solved with an algorithm without nested loops?
You can write an interpreter for any reasonable instruction set usin a single loop with a very, very long if/ else if / else if... statement. That should cover about all solvable problems. You can calculate the Ackerman function with one loop without recursion (not in the lifetime of the universe for n >= 4, but in principle).