We show that there are vast families of contact 3-manifolds each member of which admits infinitely many Stein fillings with arbitrarily large Euler characteristics and arbitrarily small signature— disproving a conjecture of Stipsicz and Ozbagci. To produce our examples, we use a framework which generalizes the construction of Stein structures on allowable Lefschetz fibrations over the 2-disk to those over any orientable base surface, along with the construction of contact structures via open books on 3-manifolds to spinal open books introduced in On symplectic fillings of spinal open book decompositions by S. Lisi, J. Van Horn-Morris & C. Wendl .
"Families of contact 3-manifolds with arbitrarily large Stein fillings." J. Differential Geom. 101 (3) 423 - 465, November 2015. https://doi.org/10.4310/jdg/1445518920