What exactly is vague about the statement “A number larger than every other number”?
Closure
If you add two real numbers, you will get another real number. What makes you think that (L) is a real number and that what applies to real numbers also applies to it?
Of course, you’re aware of the possibility that (L) is not a real number. Earlier you said “So, (L) is some special type of number, it doesn’t follow normal rules of arithmetic. So it’s possible that (L-1 = L), we can’t really say because we don’t really have a definition of what (L) is.” But we do have the definition of (L). You’re merely choosing to ignore this for some reason.
Associativity
Earlier, you said “Right, so (L + (1-1) \neq (L+1)-1). That means (L) isn’t a real number, because addition and multiplication aren’t associative on it.” I never confirmed or denied this inequality since it doesn’t strike me as particularly necessary. What makes you think that (L + (1 - 1) \neq (L + 1) - 1) can’t be the case?
I’m now going to return to this:
What about (\sqrt{-1})? Is it undefined? Is there a real number the square of which is equal to (-1)? If the answer is “no”, does that mean you reject complex numbers? If you do not reject complex numbers, why do you reject pseudo-numbers such as numbers larger than the largest number? And why do we even have to speak of such numbers?