How many years did Koreans wish their kings to live?

The answer: 1,000 years

Koreans would wish their kings a long life by raising their hands to the sky and shouting, "천세" (千歲) three times. 천세 means "a thousand (천) years (세)." At ceremonies a high ranking Korean official would lead the cheer by saying the following:

山(산)呼(호)千(천)歲(세), 山(산)呼(호)千(천)歲(세), 再(재)山(산)呼(호)千(천)千(천)歲(세).

[To] the mountain (山) shout (呼) a thousand (千) years (歲);
[To] the mountain (山) shout (呼) a thousand (千) years (歲);
Again (再), [to] the mountain (山) shout (呼) a thousand (千) thousand (千) years (歲).

山 (산) means "mountain," but here it seems to be used as a metaphor for the king or emperor. Notice the third cheer was "a thousand (千) thousand (千) years (歲)" instead of just "a thousand (千) years (歲)." A "thousand thousand years" means "a million years." The response to each prompt was simply "천세," except for the third one, to which the response was "천천세" (千千歲).

Why did Koreans say "a thousand years" instead of "ten thousand years"? [The Korean word for "ten thousand years" is 만세 (萬歲).] The reason was that Korea was a vassal state of China, and the "ten thousand year" cheer was reserved for the emperor of China.

On the 12th day of the 10th month in 1897, King Kojong of Korea was officially crowned "Emperor Kojong." At the ceremony, immediately after crowning the emperor, State Council (의정부) Minister (대신) Sim Sun-taek (심순택 - 沈舜澤) led the following cheer:

山(산)呼(호)萬(만)歲(세), 山(산)呼(호)萬(만)歲(세), 再(재)山(산)呼(호)萬(만)萬(만)歲(세).

[To] the mountain (山) shout (呼) ten thousand (萬) years (歲);
[To] the mountain (山) shout (呼) ten thousand (萬) years (歲);
Again (再), [to] the mountain (山) shout (呼) ten thousand (萬) ten thousand (萬) years (歲). 

The response: "만세, 만세, 만만세."

Notice that after King Kojong became Emperor Kojong, the chant changed from "one thousand years" to "ten thousand years." Also, notice the 만만세 (萬萬歲), which means "ten thousand (만) ten thousand (만) years (세)." That adds up to "a hundred million years."