What is the weight at which an aircraft no longer requires a licensed pilot on board in Europe?
The aircraft weight needs to be 70 kg or less. FlyNano mentioned this on their FAQs page (my paraphrasing):


  Do I need a pilot's license to operate FlyNano?
  
  According to the European Aviation Safety Association regulation
  ... FlyNano is below the necessary weight of 70kg ... In Finland for
  example no classification, no aviation license or medical
  certification are needed. But skill is needed. Nevertheless a good
  understanding of water plane operation, rules and aviation safety is
  requested. ...


I found that any aircraft less than 70kg is not affected by EASA regulations. But regardless, and emphasized by FlyNano also, pilot skills are required.