What's the best and fastest drug that builds platelets if the patient will not take any blood products?

Depends on the cause: Thrombocytopenia (low platelets) can be caused by a variety of conditions. They can be consumed (immune-mediated thrombocytopenic purpura (itp)), or the body can't produce them (leukemia or aplastic). For itp, steroids or eltrombopag are therapies that do not come from blood. Ivig is typically used for itp, but is collected from donors (i don't know if this counts as "blood products").