John Conway's account of his "Fairy Tale Platonism" was very interesting. "Fairy tale" because mathematics is not about things in the world, "Platonism" because there is an objectivity involved.
A theme emerging in recent philosophy of mathematics is the idea that we may have conflated two dimensions with our debates concerning realism. On one dimension you wonder about the mode of existence of mathematical entities (concrete like a chair, abstract like democracy, fictional like Oliver Twist, etc.). The other dimension concerns what it is that constrains the mathematician in her choice of research. Is there something beyond logic, but short of mere fashionability which dictates that some concepts just fly after they've been introduced (e.g., quantum groups) while others never really make it?
In chapter 9 of my book (see website below), I discuss an argument between mathematicians as to whether the groupoid concept is a "natural" one. Points made for groupoids talk of the inevitability of their appearance (occur in many fields independently), their simplicity (especially in category theoretic language), the avoidance of arbitrary choices when they are used. The opposition might admit their mild convenience, but take groups to be the real McCoy when it comes to symmetry measurement.
As for myself, I find debates aimed at this second dimension far more interesting. To what extent are mathematicians constrained by their psychology and the tastes of their community? Don't the surprising interconnections achieved by some concepts (e.g., quantum groups to knot theory) suggest a "getting things objectively right"? What should we make of the often very distant empirical sources of mathematical concepts?
In the case of surreal numbers, once the generating mechanism of gap-filling is in place presumably matters are fairly forced. But before this, in the "space" of such mechanisms is there something that singles out the gap-filling one as especially interesting?