You're right! あきらめかけて is a compound verb from あきらめる and かける. あきらめかける←あきらめ（る）＋かける verb + かける means just before an action or event and verb + かけた is not only a past tense of verb + かける but also carrying a connotation that it didn't happen actually. So it means pretty much like 'almost' or 'nearly', I guess. ほとんどあきらめかけてしまった means that I was almost ready to give up. ( but I didn't give up actually.) 彼に何か言いかけたが、彼は行ってしまった。 I almost started to saying something to him, but he had gone. (Actually I didn't say anything to him.) 酒を飲みかけたが、やめた。 I was almost about to drink alcohol, but I stopped it. (Actually I didn't drink any.) 死にかける。 I was about to die. (Actually I didn't die.) I hope this helps you!

Hello, Hilary-san. Hello Selly-san. I read this article and was interested a lot to see the link Hilary-san gave us. There, I found the reason why Hilary-san could not find あきらめかける on the database. I had never known OUR compound verbs can be categorized into two kinds. Please try the following link and click the “Syntactic and Lexical Compound Verbs” at the column in the left side of the page: 【 http://vvlexicon.ninjal.ac.jp/en/ 】You’ll see the reason and V-かける at the top of the list of the compound verbs excluded from the database. Now, Hilary-san, you are sure to combine あきらめる or almost any other verbs and かける.