Postzeep: Your question is interesting even if the universe is infinite...let's set that aside for a moment...[There are different size infinities.]
I can think of a few things to consider about relative sizes of universes...mostly initial expansion [INFLATION] and subsequent, slower, expansion. We'll restrict ourselves to descriptions based on general relativity: The general model for that is FLRW model but 'distances' are not so simple...In general more mass in our era would mean slower expansion...but it's not so simple, I don't think.
I'm guessing the 'biggest universe is the one in which both periods have the fastest growth...the most acceleration...and retain the most. Or one in which inflation never ends. Inflation had to be 'glued on' to explain things we observe; then 'slow roll' had to be glued to inflation so it would stop! then we add a cosmological model! That's homework we currently model Gr based universes.
Here is how Roger Penrose describes the initial inflationary period, right after the big bang from THE ROAD TO REALITY, pages of section 28.4. I am paraphrasing here:
The commonly described history of the universe begins with conventional causal expansion. This means, for example, that thermodynamic energy had enough time to create rather uniform temperatures...to 'communicate' causal effects. Then from about 10-35 second to about 10-32 seconds an arbitrary scalar field is introduced [sometimes referred to a 'Higgs field']. 'Some models require more than one inflationary phase, in which case there would have to be a different scalar field for each'.
...'During the inflation stage we have a region of a false vacuum which represents a quantum mechanical phase transition to a vacuum different from the one we are familiar with today'...Lambda [the cosmological constant of expansion] was about 10 110 times it's current density.
This transition is the 'slow roll' version described in Wikipedia...
http://en.wikipedia.org/wiki/Cosmic_...roll_inflation The initial inflationary expansion is theoretically driven by a negative-pressure vacuum energy density...a Higgs field...so if you want more rapid expansion, check out what increases that negative pressure vacuum energy. And once it is a big as you can make it, eliminate the 'slow roll' which brings it to an end...and it will continue forever. Of course there won't be much of interest in such a universe; certainly no life forms, for example. You can also check 'vacuum expectation value' to learn more about vacuum energy.
The FLRW cosmological model starts moments after the inflationary period as I recall...I think that means if you want to know anything about relative sizes of universes, you'd have to decide what makes inflation the fastest, then subsequently, what keeps expansion moving the fastest...
You can get a snapshot of some differences here:
Use the black curve which is one distance measure, the Hubble distance.
http://www.astro.ucla.edu/~wright/cosmo_02.htm
scroll down the page until you come to three side by side charts across the page...
H versus Z...
but this is for a rather limited period of time...