Theory:

     This, however, was not my worst experience on a plane flight. My worst experience was when I was writing important thoughts in a notebook (‘buy socks’, ‘clutch drinks carefully’, etc.), sucking thoughtfully on the end of my pen as you do, and fell into conversation with an attractive young lady in the next seat. I amused her for perhaps 20 minutes with a scattering of urbane bons mots, then retired to the lavatory where I discovered that the pen had leaked and that my mouth, chin, tongue, teeth and gums were now a striking, scrub-resistant navy blue, and would remain so for several days.
Explanation:
 
Following the two incidents that had happened during a flight, the narrator proceeds to the lesson’s \(4th\) narrative. The narrator creates anticipation in the audience by saying that the previous two incidents, though quite embarrassing and terrible in nature, were not as terrible as the following incident. He claims that the incident he was about to narrate had been the worst experience of his.

The incident, as expected, had also taken place on a flight. The narrator was writing important notes such as 'buy socks' and 'clutch drinks carefully' on his notebook. It is funny how the narrator calls these seemingly mundane tasks as "important". The irony here is that though the tasks may sound petite, the narrator actually considers them important and high priority.
 
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The narrator was writing important thoughts in a notebook

So, back to the incident, the narrator was thinking hard of the important things to be noted down and was sucking thoughtfully on the end of his pen. Meanwhile, he fell into a conversation with an attractive young lady in the next seat. He indulged in an exciting conversion, throwing witty remarks occasionally, for about \(20\) minutes. Later, when the narrator excused himself to the lavatory, he was startled to find that the pen he had been sucking on had leaked. His mouth, tongue, teeth, gums, and even his chin had turned navy blue. Moreover, the ink was very bright and scrub-resistant. Despite trying to rub it off, it remained on his face for several days.

Poor narrator! It is worth to imagine how the narrator must have faced the lady, now that he knew his face had been ‘blue' during the entire conversation. Or rather (to begin with), how and 'when' he had left the lavatory, given that the stain was stubborn and scrub-resistant.
 
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The narrator's face had blue stains
 
Meanings of difficult words from the paragraph:
 
S.No
Words
Meanings
1
Bons motsA witty remark
2
Scrub-resistantA stain that is tough and hard to remove
Reference:
National Council of Educational Research and Training (2006). Moments. Accidental Tourist- Bill Bryson (pp. 56-59). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.