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This is the last lesson about important nominal phrases. Most of them are related to each other, so that will make things easier. 

As a noun, 至極 shows an "extremity" of something. As an adverb, it means "extremely" or "exceedingly".

1. 至極便利な電気器具ですね。
    This is an extremely convenient electric appliance, isn't it?

2a. 滑稽至極
2b. 滑稽の極み （More natural)
      The extremity of ridicule

3a. 恐縮至極
3b. 恥ずかしさの極み　（More natural)
      The extremity of shame

4. 至極当然である。
    To be exceedingly obvious.

極み is a noun meaning "sublime", "extremity", and "height" and shows the utmost. This word is typically not used in speaking. 

5. 絶望の極みに陥(おちい)る。
    To fall into the height of despair. 

6. 興奮の極み
    The utmost of excitement

7. ここで断念するのは遺憾(いかん)の極みだ。
    Abandoning here is the height of regret. 

8. 贅沢ぜいたくの極みの生活をする。
    To live a lifestyle of utmost luxury.

至り either means "the utmost" or "result". 

9. 不始末ふしまつは若気わかげの至りと反省はんせいしております。
    （物事において）不始末であったことは、私の若気の至りです。
     Carelessness is reflecting on the stupidity of my youth. 

10. ご同慶(どうけい)の至りに存じます。
       I offer you mutual congratulations.  

11.恐縮きょうしゅくの至りです。
      I am sorry to the utmost.

万一 expresses that something rarely happens, but there is a possibility of the otherwise. It may be translated as "by some chance/possibility". It may also be used as an adverb. 

12. 彼に万一のことがあれば
      If anything happens to him

13. 我々は万一を考えなければならぬ。  
      We must think and prepare for the worst.

14. 彼は万一の場合に備えた。
       He prepared for contingencies.

15. 万一遅れたら、先に行ってくださいね。
      If there is the chance that you could be late, please go earlier than (planned), alright?

ゆかり, in 漢字 as 縁, means "affinity", but it can also show that something or someone is related to some place. It is this usage that catch people off guard. 

16. 漱石そうせきゆかりの地
      Place in connection with Soseki 

17. 文豪ぶんごうゆかりの地
      Place in connection with a literary legend