What is 多【おお】き?
In brief, this is the Classical Japanese attributive form of modern 多【おお】い. The attributive form or 連体形【れんたいけい】 is the specific conjugation form used when an adjective or verb is used to modify a noun or noun phrase.
Deeper dive: What is this -ki ending?
All modern -i adjectives have the same -i ending for both the 終止形【しゅうしけい】 ("terminal / conclusive form", the form used to end a sentence) and the 連体形【れんたいけい】 ("attributive form", the form used to modify a noun or noun phrase).
Examples:
- 車【くるま】が小【ちい】さい -- terminal
- 小【ちい】さい車【くるま】 -- attributive
- 車【くるま】が新【あたら】しい -- terminal
- 新【あたら】しい車【くるま】 -- attributive
In Classical Japanese and older stages of the language, the terminal and attributive forms were different: the terminal ended in ~し, and the attributive ended in ~き.
As an extra wrinkle, adjectives with modern endings of just ~い (so-called ク活用【かつよう】形容詞【けいようし】, or "-ku conjugation adjectives", named for the distinct adverbial ending, the same for both modern and Classical) conjugated slightly differently from adjectives with modern endings of ~しい (so-called シク活用【かつよう】形容詞【けいようし】, or "-shiku conjugation adjectives"). The key difference is that the terminal for -ku adjectives replaces the ~き with ~し, while the terminal for -shiku adjectives just drops the ~き (so you don't have doubled ~しし).
Examples:
- 車【くるま】が小【ちい】さし -- terminal
- 小【ちい】さき車【くるま】 -- attributive
- 車【くるま】が新【あたら】し -- terminal
- 新【あたら】しき車【くるま】 -- attributive