[jhpriestley]

I only took a couple courses in japanese, but I'm pretty sure that this quote vastly overstates the difficulty of politeness levels for verbs.

Each verb has different forms for formal/informal/very formal, but they're highly regular. For example "iku" is "go" and "ikimasu" is "go (polite)". "wakaru" is "understand" and "wakarimasu" is "understand (polite)". It's not like you have to learn four forms of each verb, you learn the basic one and then you know the rest. The ultra-formal versions, which are more rarely used, are also regular - they involve the infinitive of the verb plus a conjugated auxiliary verb IIRC, like saying "presume to go", "presume to understand" etc.

[Arnavion]

I believe Feynman's quote is not referring to politeness of the "miru" verb but of using different verbs entirely. For example meeting someone is both "au" and "(o)me ni kakaru" with similar difference of formality.

[Nadya]

There is Polite Language (丁寧語), Respectful Language (尊敬語), and humble Language (謙譲語).

Compare:

見る -> 見ます -> ご覧になる -> 拝見する

That is a lot different than just learning four different conjugations for 見る - it becomes an entirely different word!