愈々（いよいよ）, seen primarily in hiragana, is an adverb that indicates that (A) has or will ‘finally happen’. In many cases this is after some state that has been continuing for a long time, or has been compounding further and further until (A).
Like many other sentence altering adverbs, いよいよ will appear at the beginning of a phrase. It is also considered quite casual when compared to other grammar points like ついに.

• いよいよ明日（あした）で卒業（そつぎょう）か...。
  Tomorrow is finally our graduation, isn't it…
• いよいよ俺（おれ）の番（ばん）だ。緊張（きんちょう）するけど頑張（がんば）らなくちゃ！
  It is finally my turn. I am nervous, but I must do my best!

As いよいよ highlights something that is compounding, it can also frequently be translated as ‘more and more’.

• 台風（たいふう）が接近（せっきん）するにつれて、いよいよ風（かぜ）が強（つよ）くなってきた。
  As the typhoon approached, the wind started to blow more and more.
• 彼（かれ）の話（はなし）を聞（き）いていると彼（かれ）がいよいよ怪（あや）しくなってきた。
  Listening to his story makes him more and more suspicious.

The difference between the nuance of いよいよ being ‘finally’ or ‘more and more’ will usually be clear due to the statement following it being either conclusive (finally), or inconclusive (more and more).