If I understand well the number you have in mind, 2^2^2^2^2^2, it is not all that large: 2 2^2 = 4 4^2 = 16 16^2=256 256^2=65,536 65,536^2=4,294,967,296 The person you heard this from may have had another number in mind, namely: 2^(2^(2^(2^(2^2)))). Let's construct this one: 2 2^2 = 4 2^4 = 16 2^16 = 65,536 2^65,536 = ??? .... and this fifth step (bringing in the fifth "2") already goes beyond most ordinary spreadsheets and calculators. Still, since 2^10 is about 10^3, we can estimate the result to be around 10^19661, i.e. a "1" with nearly 20,000 zeros. A good computer could probably do the calculation and could print out the resulting number on perhaps 12 pages or so. The sixth step, taking 2 to the power of this number, would really go beyond what most of us can even imagine. It would bring us to a number -- let's call it "K" in your honor -- that, written in the decimal notation, would have so many digits that this number of...

I think the terms in your first question are generally used in a sense that's relative to our (human) knowledge. But this need not mean that this use reflects our ignorance of causes. For there may be real chance and randomness in nature (here the words "real" and "in nature" indicate that "chance" and"randomness" are used in their more unusual sense). The currently accepted view in physics holds that this is in fact the case at least in regard to subatomic particles. The word "infinite" is generally used to refer to what really is infinite, mostly things in mathematics and geometry (the set of all natural numbers, Euclidean space). The mere fact that we don't know whether a thing has limits does not justify calling it infinite in any normal sense.