She has faced almost _____ difficulty in arranging the funds for the event.

The adverb 'almost' is used to modify words that suggest a high degree or limit. 'Insurmountable' means impossible to overcome, which fits the modifier 'almost' to mean 'nearly impossible to overcome'. 'Big', 'gigantic', and 'large' are general size adjectives that don't pair as naturally with 'almost' in this context of difficulty. Hence, 'insurmountable' is the most appropriate choice.