Another compactification that's been studied a bit is the Satake compactification.  Unlike the Thurston compactification, this is definitely algebraic geometry!

Satake introduced a natural compactification of the moduli space A(g) of principally polarized abelian varieties in genus g.  Later, Walter Baily proved that this compactification turns A(g) into a projective variety.  The construction uses theta functions and can be read about in Igusa's book "Theta Functions".

Anyway, let M(g) be the moduli space of curves.  The map that takes an algebraic curve to its Jacobian induces a map M(g)-->A(g), and Torelli's theorem says that it is injective.  Baily later showed that the closure of the image of M(g) in the Satake compactification of A(g) is also a projective variety compactifying M(g).  Incidentally, this provided the first proof that M(g) is a quasiprojective variety!