アテジの例 (ateji)。

Just happened to be looking up めでたい on jisho.org, and here's what came up:

目出度い
愛でたい
芽出度い

Granted, for each entry it says that the word is usually written with kana alone. But how interesting, that there seem to be no kanji associated with this word beyond sound—all examples seem to be jury-rigged to fit the phonetics. In the first, the kanji have readings of me (moku, an eye), de (deru/dasu, exiting), and tabi (do, a "time"/repetition). Me・de・ta・i. The second uses ai 愛, love, as the "m" component; Saiga turns up only two entries in which 愛 carries readings starting with "m", both cases asterisked:

めでる, mederu, to appreciate (eg, beauty) in a loving way: 花を愛でる
まな mana, beloved, as in 愛娘, beloved daughter

These really seem to be one-offs, 愛 forced into contexts in which it's not comfortable.

The third example in the above list uses 芽, め　me, a sprout, for that sound. め・で・た・い.

So presumably when kanji were being fitted onto spoken Japanese, nobody came up with anything good for めでたい, and at some point someone stuck on 芽, 目, 出, 度, etc., because there ought to be something. Or, maybe there was a very good explanation at the time, which now either has been forgotten or is perfectly clear to people who understand Japanese much better than I do. Part of the adventure.

Speaking of: I still don't get the -ou and -i forms: めでたい, おめでとう;　はやい, おはよう, etc. Interesting that the form that follows the -ou is not です(である), but just ある—not でございます but just ございます. So, 「おはようーさんです!」, but 「おはよう...ございます」.