On the fundamental group of the complement of two tangent conics and an   arbitrary number of tangent lines
  We compute the fundamental groups of the complements of the family of real conic-line arrangements with up to two conics which are tangent to each other at two points, with an arbitrary number of tangent lines to both conics. All the resulting groups turn out to be big. 
Meirav Amram, David Garber and Mina Teicher