My knowledge of world languages is nowhere near sufficient to know whether there exist any purely additive or purely multiplicative natural languages, so I'll use some particular instances from two languages as examples. Latin is kind of the classic multiplicative-complexity language (although some parts of it have additive complexity); we are going to talk about the Latin close demonstrative pronoun ("this/these" in English).
As I mentioned quite some time ago, Latin inflects nouns based on gender (masculine, feminine, and neuter), number (singular and plural), and case (nominative, genitive, dative, accusative, and ablative). This results in their being 30 different "this/these" pronouns (!), as shown (I'm gonna do my best to make this look okay without FrontPage - what I usually use when doing complex formatting):
Nom Gen Dat Acc Abl
Masc. hic huius huic hunc hoc
Fem. haec huius huic hanc hac
Neut. hoe huius huic hoc hoc
Masc. hi horum his hos his
Fem. hae harum his has his
Neut. haec horum his haec his
Ugh. That's disgusting. As you can see, there is some degree of regularity, but enough exceptions that you really need to memorize about 18 of the 30. This is what I mean by multiplicative: the number of different forms of a given thing that must be memorized are roughly equal to the multiplicative total of all the different ways in which the thing can be inflected (3 * 2 * 5 - gender, number, case - in this example).
English is a bit better than Latin, although that's partly due to the fact that, compared to Latin, English inflects very few of its words, and those it does inflect have few variations. Japanese, however, has an example of additive complexity that's just beautiful - its demonstrative and interrogative pronoun system. Words are inflected by class (specifically, how far away the thing being referred to is) and form (whether it's a noun form, adjective form, etc.) as shown:
Near Far Further Int.
Noun kore sore are dore
Adj. kono sono ano dono
Example konna sonna anna donna
Manner koo soo aa doo
Place koko soko asoko doko
In case it's not obvious, the English translation of the first column would be: "this thing" (noun), "this" (adjective), "such as this" (example), "like this" (manner), and "here" (place).
Thus, knowing only the four distance class prefixes and five word form suffixes, we can form all 20 combinations while only having to remember a single irregularity - asoko ("way over there" or some such). This is taken even further with indefinite pronouns, in which further suffixes are added to the interrogative forms above (we'll use "doko" - "where" - as an example), for forms such as "somewhere", "nowhere", "anywhere", "everywhere".
Additive >> multiplicative. Especially in Caia, where I intend to encode a LOT of information into pronouns and conjugation islands.