In my text I read the sentence:
あなたが体験したよりもっと[巨]{おお}きく、もっと深くさせてくれる力があるからだ。
(furigana is from the text)
I understand 大きく, and I understand some words with 巨 like 巨大. However, when I looked up 巨きく in a dictionary, I was unable to find anything.
Why do these kind of readings exist; however, they are not in a dictionary?
Basic furigana means, for 絶対{ぜったい}, 'zettai' is how you read 絶対...period.
For the non-standard cases, think of it basically the same way, but with a little twist: for 巨{おお}きく, 'oo' is how I want you to read 巨.
As a deeper example...
泥棒{おまえ}は信用{しんよう}出来{でき}ない。
If this is a line of dialogue, the person is saying "I can't trust you", but the implication is that "you" are a thief, and that is why you cannot be trusted.
When the furigana are not standard, in my experience, the furigana is what is said, and the kanji is what is meant. For your example, there's not a lot of difference between 巨 and おお(きい), but it should be basically the same thing: おお is how it's said, but 巨 is the underlying meaning.
I would say, your question is "not well posed". When you study Japanese, you are taught that each kanji has readings. It's taught that way, because, I presume, it's easier to teach that way.
A more precise interpretation of what's happening in Japanese is that you choose a kanji for the Japanese word. In practice, there are a few very common choices for each Japanese word; so much so that it appears as if each kanji had fixed readings associated with it.
That's why you thought that the association of 巨 with おおきい was invalid. It's not invalid. It's just that the writer chose the kanji for the Japanese word おおきい most likely because the kanji best expresses the nuance she or he wanted.
The dictionary doesn't list this association of 巨 with おおきい simply because it's not common. The dictionary doesn't set rules on what kanji can be used for what Japanese words; it simply records most common associations between kanjis and Japanese words.
Edit: I've come across this use of 巨: 「それは異様に頭の巨きい少年であった」from "Hanakatami (花筐)" (1937) by Dan Kazuo (檀一雄)。There is one more instance down the same page. At the first occurrence, there is a furigana in my copy but I don't know whether it was added by an editor. For this case, the furigana isn't necessary because the context and the きい part of 巨きい unambiguously imply its reading.
. . . The point I'd like to make here is that the simplistic view that each kanji has a few fixed readings is a product of the simplified, standardized education after World War II in Japan. Old writers, as well as good writers today, don't take that view. They just use whatever kanji is "best".
Below Ci3 asks a perfectly valid question: [to interpret his/her question in my own way] How can one then decide what's a good kanji for each Japanese word? Even though it's a good question, there is no good answer to it. Suppose you are writing English and trying to describe a color. You wonder whether you should write "red", "deep red", "scarlet", or "red-purple". Which is best? There is no answer to such a question. You can safely say that "green" is an extremely bad choice to describe the color and you can also say that "red" is always the safest choice although it may not best describe the color you are looking at. . . . Same idea. Although 大 is always the safest choice for おおきい, it may not be the best one for the instance.
Though different from the example you gave, there are examples where the furigana even contradicts the okurigana(!!)
For example, the word [崇まふ]{あがまう} - the furigana there gives the pronunciation of the entire word - has a spelling which ends in ふ, but the pronunciation ends with う. Another example is [目合ひ]{まぐわい} - there is no ひ at the end of the pronunciation.†
As for "why?" - both of these examples, along with all the others I could find, are considered archaisms, from before the language reforms of the mid-1900's. They're not something you'd run into often. I have no idea if they're even commonly known to native speakers (I only encountered them when they broke the okurigana logic in a dictionary parser I worked on)
† For anyone who disagrees: both the English-Japanese database JMDict and the Japanese dictionary 広辞苑 agree that these are the correct pronunciations
