Hypothetically, 6 or more phase line with same mechanical energy?
Let's assume I have a 3-phase generator and need X amount of mechanical power in order to get 200 Volts Phase-Neutral. 


You are confusing the relationship between power and voltage. For a given winding arrangement the voltage will depend on the speed. The mechanical power input required when unloaded will be that only to overcome the mechanical resistance.


  Now suppose I add some windings etc. so that I have a hypothetical 6-phase system. Would I need more mechanical energy in order to get 200 V for each phase, or can I get it by same amount X?


You don't need much power to generate voltage. You do need power to generate current at that voltage.


  So basically, if I have more phases do I need more mechanical power?


No. You need more mechanical power if you draw more electrical power from the generator no matter how many phases you have. In general:

$$ P_{OUT} = P_{IN} \times efficiency $$