[giberson]

     What do you mean, 'infinite in dimension'? The universe has only 4 dimensions, 3 spatial, 1 in time.

I'm was not asking how many dimensions the universe has. I was asking if the universe is infinitely tall, infinitely wide and infinitely long.

     Why do you need to invoke infinite speeds and times? 

Simply for matters of scale. I was trying to avoid considerations like "it's so big, it would take so long ...".

     If 2 particles stay in the same place [...]

Actually, it's one particle that moves, and other is stationary, thus providing two points of reference for determining a displacement. Specifically the intention was to indicate a path through space such that you always are moving further from where you started and will never arrive back at your starting point (as opposed to the spherical theory of space such that you can only go half way around the sphere until you start getting closer to the point at which you started.

      If you pick a random direction and travel really fast for a really long time, the chances of you running into any other matter is extremely remote. 

I didn't mean physically collide with other matter, I simply meant to pass by it in the "vicinity". Similar to how traveling along a highway you pass by or through towns, you don't physically collide with them though. If matter in the universe was finite, then regardless of how sparse that matter is, there's an imaginary bounding box you could draw around all matter in the universe. A particle moving really fast for a long time would eventually end up outside of that bounding box. At which point its distance not just from the stationary particle at it's original, but all particles of matter would forever be increasing. However, if matter is in fact infinite, then such a scenario can not occur.

Luckily, despite my questions barely making any sense, you've managed to provide in your reference [1] a comment that answers exactly my questions so thank you for that.