@Mark: Yeah, if you mean in a positive way. In a negative way, you can use 하마터면. For example, we say "간 떨어질 뻔했네" when someone or something(like a fierce dog barking) just scared us. 거의 간 떨어질 뻔했네 sounds awkward, whereas 하마터면 간 떨어질 뻔했네 sounds natural and more emphasized than the original phrase. 하마터면 앞차랑 부딪힐 뻔했어요. I almost bumped into the back of the car ahead. 하마터면 큰 일 날 뻔했어요. I almost got in a world of trouble. And Emily, I realize I'd unconsiously translated every "almost" of your examples in my previous answer.

Yeah, Mark is correct. Consonant ending verb : -을 뻔하다 Vowel ending verb : -ㄹ 뻔하다 I almost believed you. 거의 믿을 뻔했어요. I almost fell over laughing. 웃다가 거의 쓰러질 뻔했어요. during the meeting i almost started laughing but thankfully i held back 그 모임에서 거의 웃을 뻔했는데 어떻게 잘 참았어요. i ate almost all the cookies 쿠키 거의 다 먹었어요.

thanks Hailey^^~ am I correct in saying that 그의 adds emphasis in the meaning with 뻔 하다?

~ㄹ/을 뻔 하다 or 거의 i almost believed you ~ 믿을 뻔 해요. i almost fell over laughing ~ 웃다가에서 거의 떨어졌어요. (you could also use 쓰러졌다). the other 2 sentences, I would paraphrase them. *I'm still learning also, so don't quote what I have written in 한국말 ok? One of the usual suspects will do a better job^^