In which stage should a company start applying an ITIL plan?

At anytime... you can start to follow best practices from the beginning. You just have to ensure that your processes are well described in your operation manual, they should be also communicated to all, understood and applied by all.  Then Cobit or ITIL can be applied. You just have to ensure the respect of the processes at all levels. Do it simple and harden it step by step.Service Design, Service Transition, Service Operation, Continual service improvement, and their functions Lifecycle Management, Configuration & Asset Management, Service Delivery, Incident, Change, Problem Management are the basics of the production management.It's easier to start at the beginning to respect best practices than after. Think about the inertia and difficulties to implement ITIL you will encounter due to usual accumulated discrepancies and lack of process improvement.