Learn English – Why are there multiple prefixes for the same root word

^{I’m gonna have to disagree with @prash, and so I’ll post this as a competing answer, offering an alternative take on The Rule:}

1. The prefix mono- comes from Greek monos, itself rooted in the Proto-Indo European *men- (small). uni- comes from the Latin unus, itself from PIE oinos (and thus related to the Greek oinos, “ace on a dice”; Ancient Greek otherwise used alpha for the number one).

2. Forming words from mixed Latin and Greek roots used to be frowned upon. In the classical tradition, you would form things either from Greek or Latin, not mix them together.

3. As time evolved, a number of Greek and Latin roots entered the English lexicon, mostly as suffixes and prefixes. Because they became part of the language in that way, they could be combined with other words, being somewhat free of the older rules: you wouldn’t go forging a brand new words directly from Latin and Greek roots, but you would add a prefix (say, auto) that came from Greek but was now assimilated into the English language, and put it in front of an English adjective (say, mobile, which came from Latin movere through Middle French). And kaboom, you have: automobile. And few frown upon it.

4. So, what's the current status? It's now mostly a matter of taste. There is no strict rule, and you should form the word that sounds better to your ears.

Wikipedia calls words formed with roots of different languages hybrid words, though I've never really heard the term used. I’d prefer we call them heteroradicals (tip of the hat to JSBᾶngs).

Moreover, the Wikipedia article linked above has a long list of such English words, many of which are very common: homosexual, television, hyperactive, electrocution, bigamy, Minneapolis, etc. It even includes the following two words which have the mono- prefix:

• monoculture
• monolingual

TLDR — I usually don't do tldr’s, but this has turned into a general post on the topic while you had a very specific question. So, here's the answer to your very specific point: it's a matter of taste, choose the one that sounds best. If the roots are from the same language, as an added bonus it will please some more people.

After some digging, I think there is a term that can satisfy all the varied thoughts and criticism thus far. A word having two (or more) prefixes incorporated is an:

Augmentation

Augmentation is modification of a word by any of the language's related forms (prefixing, suffixing, etc).

The OED has this definition for an Augmentitive, adj. and n.:

2.b. Grammar. Of a word: augmenting the properties of the term whence it is derived, or generally expressing augmentation of an idea. (Augmentative words are generally formed by the addition of augmentative affixes.)

Emphasis mine.

This solidly establishes that Augmentation is a modification that can involve two or more modifiers, including Prefixes.

We can see the concept of augmentation being used in the grammatical context of prefixing, while also being distinguished from (and I would say not limited to) a prefix. This reinforces the OED's definition.

1. 1646 Sir T. Browne Pseudodoxia epidemica iii. xxiv. 170. The Greek word Bous, which is a prefixe of augmentation to many words in that language.

This establishes that "a prefixe" (a single one) is a kind of augmentation.

2. 1671 E. Phillips New World of Words (new ed.). The Syllabical Augment, is an augmentation which is made in Greek verbs, by prefixing ἐ (and thereby adding one syllable).

This establishes that a syllabical augment is a kind of augmentation.

This use of Augmentation also follows the general usage of the word, which is not limited to a single [dimension] of modification. Cf. those in Heraldry and Pharmacology.

I originally said that, No, there is no specific term for this kind of word prefixing. I think that some may still find route to argue that position. If we need a term, I don't see us doing better than this.