So, if you will promise not to throw pies at me, I will suggest that the universe may in fact have an arbitrarily large, increasing, but finite mass.
Since there are very distant massive objects whose speed relative to us is close to that of light, their relativistic mass must be approaching inifinity. Since the universe's expansion is accelerating, we can only expect this mass to increase without bound.
Good thing that stuff is far away.
-Erasmus