What does "nearly infinite mass" mean? I am sure this is a silly question, but I was reading something that described the pre big-bang universe as having "nearly infinite mass."
How can something be "nearly" infinite?  The term seems to make no sense. 
 A: Any natural number, no matter how large, is "infinitely far away" from infinity, i.e., there are an infinity of numbers larger.  So, your intuition is correct.  "Nearly infinite" seems very odd indeed.
Of course, "pre big-bang universe" sounds suspicious too as does the notion that whatever that was has the property of mass. 
A: It sounds like the article is not credible. They could have easily used "Had a mass approaching infinity". Which would mean a non-static ever increasing mass. I suggest seeking reading material that is properly referenced and supported with mathematically rigorous explanations.  
Anyone can post opinionated "ideas", but without proper references all it is is wild conjecture.
A: The universe seems to be well described by a solution to Einsteins equation called the FLRW metric. If you extrapolate this backwards towards the Big Bang the density of matter increases, and at the Big Bang itself the density becomes infinite.
A quick side note: the universe may well be of infinite size, so it's mass is necessarily infinite. It only makes sense to talk about the density i.e. the amount of matter in some chosen volume.
Anyhow, few believe that the density actually becomes infinite. It's widely believed that quantum gravity, or something like it, intervenes as we approach the moment of the Big Bang. I'd guess that the article you've read means that the density becomes extraordinarily large as we approach the Big Bang, but never becomes infinite. Hence the description "nearly infinite". However I agree with the other answers that the phrase "nearly infinite" is meaningless outside of a metaphor.
