1) Is there any item in the game that has the Fortify Enchanting effect?

2) What is the recipe for a Fortify Enchanting potion?

1) No - there is no Fortify Enchanting enchantment.

2) Any combination of Blue butterfly wing, Hagraven claw, Snowberries or Spriggan sap

Does the # of ingredients have any effect on the strength of the fortify potion?

I don't believe so, no. And I should add that my ingredients list is only for vanilla. If you have DLCs there are 3 other ingredients for it, I think.