This is a great question. It is best to start off with the observation that fear permeates the whole book. Therefore, it is not possible to speak of the greatest fear. Fear is all around. Let me give you a few examples.
The Franks hear from the outside world that it is getting worse and that Jews are being found. So, as each doorbell rings or there is a knock on the door, their fear grows and grips them. We can say that they fear detection from the Nazi's and separation from each other and death.
They also hear that things like food are becoming scare. So, they fear for their own physical health in the annex.
Anne also fears loneliness at life in the compound. There is a constant feeling of anxiety and no one want to burden another and they keep all this fear within them. For this reason, they are all on edge. We can say that there is social fear.
Finally, Anne fears whether she will ever truly be able to be free and live her life. Here is a quote:
"It’s difficult in times like these: ideals, dreams and cherished hopes rise within us, only to be crushed by grim reality. It’s a wonder I haven’t abandoned all my ideals, they seem so absurd and impractical. Yet I cling to them because I still believe, in spite of everything, that people are truly good at heart."
In the end, fear is embedded in to novel.