Big numbers

Japanese has had the good fortune of inheriting the Chinese number system. The way of building higher numbers is so logical that a learner can pick it up very quickly, and it makes arithmetic much easier for children, because they are already well aware of the concepts of hundreds, tens and units by the time they start school.

Powers of ten, however, are not quite as logical. In fact, the usage of the higher powers are not universally agreed on in the kanji using world. There were 下数, using increments of 10; 中数 using increments of 10,000; and 上数 where each number name was the square of the previous. In Japan 中数 became standard in the 17 century, although apparently remnants of the different systems persist elsewhere.

From 1 to 10,000, each multiple of ten has its own name.
一 いち 1
十 じゅう 10
百 ひゃく 100
千 せん 1,000
万 まん 10,000

Then the numbers go up in multiples of 10,000.
億 おく 100,000,000 (10⁸ a hundred million)
兆 ちょう 1,000,000,000,000 (10¹² a trillion)
京 けい 10,000,000,000,000,000 (10¹⁶ ten quadrillion)

That's about the limit as far as useful numbers go, but they do go much higher!
Even from 兆 (portent, omen) and 京 (capital city) characters are being reused. There isn't always a clear explanation as to why a particular kanji was chosen, as they are most likely phonetic borrowings.

垓 がい　10²⁰ The character means boundary or limit.
禾予(as one Kanji) じょ　10²⁴ The character for this is so rare that it isn't even in the basic JIS character set. The original kanji was 秭, which means 'piling up'. When writing this, I found that I could paste the kanji in the editor, but it wouldn't display on the published post.
穣 じょう　10²⁸ Lush, abundant.
溝 こう　10³² A ditch or narrow waterway.
澗 かん　10³⁶ A mountain stream. Two water related kanji in a row. Is there a pattern coming up? Sadly, no.
正 せい　10⁴⁰ Usually means 'correct', but has a minor meaning of 'long' which could be why it is used.
載 さい　10⁴⁴ Load or carry. Nothing could possibly carry anything this big! It also means write down or print, so is more likely to have the meaning 'something so large it cannot be written'.
極 ごく　10⁴⁸ Limit, extreme. The absolute highest number ever! Hang on a minute....

After that, Buddhism was clearly a major influence.

恒河沙 ごうがしゃ 10⁵² 恒 is a kanji transcription of the Sanskrit name for the Ganges, 河 is river and 沙 means 'sand', so the whole thing is 'the number of grains of sand on the River Ganges'. Quite poetic, really.
阿僧祇 あそぎ 10⁵⁶ From the Sanskrit for 'more than can be counted'
那由他 なゆた 10⁶⁰ From the Sanskrit for 'an extremely high number'
不可思議 ふかしぎ 10⁶⁴ 不可 is 'impossible', 思議 is 'conjecture' or 'guess'. An unimaginably large number!
無量大数 むりょうたいすう 10⁶⁸ Originally two words, 無量 is 'unmeasurably huge' and 大数 is 'a big number'. So together they make an gigantastically meganormous number.

100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

 Still quite a bit smaller than a googolplexian, and slightly less than the total number of atoms in the universe.