What is an adjective for a very large negative number?

Note that large does not mean large, positive. If a number X is large, then -X is also large. Furthermore -0.00...1 is also infinitesimal.

So, no only isn't there a single word that means large, negative, but, to begin with, there also isn't one which means large, positive!

In computing, the widely used term bignum refers to any one of various data representations for arbitrary precision integers, which are not confined to machine-imposed sizes like 32 or 64 bits. Of course, bignums can be positive or negative. That is another example which shows that big is understood to refer to magnitude, not to positive displacement from zero.

Maybe you can coin some new ones to fill the void, perhaps by portmanteau: posinormous, plusantic, negormous, negantic. These may not be in the English language now, but if you succeed in popularizing them, they will be. They do exhibit English phonology and morphology.

A less radical approach would be to use some standard prefix or suffix: supernegative and superpositive. Even if nobody uses these words, the meaning is clear. In English, we can improvise new words that are not listed in dictionaries, yet which have a more or less self-evident meaning, by combining existing words with standard prefixes and suffixes. For instance, dictionaries do not have to list every single possible word that is formed by [adjective] + ness.

Your assumption that large refers to positive numbers of great magnitude is probably more correct than other answers have stated. The notions of large and small numbers derive from counting (and then measuring) and so are implicitly associated in everyday thought and language with positive numbers rather than the real numbers (as a mathematician might claim).

Imagine a "journey" from 3x10^6 to -3x10^6, it is counter-intuitive to claim that at the mid-point numbers stop getting "smaller" and begin to increase again so that the endpoint is as "large" as the beginning. The pure mathematics may be impeccable, but the man-in-the-street knows that is nonsense.

To keep the domains of mathematics and the everyday use of ordinary words in some form of alignment, I suggest that a negative number of great magnitude is one which when multiplied by -1 produces a huge result.

I have seen such numbers described as very negative or similar. I don't think there's a one-word term used in mathematics for such a number... though I suppose there might be a historical one that's no longer used.