In my prepositions chapter I found an example sentence
He sat watching far on into the night
I don't understand this bold part
On is not here a preposition, but an adverb, acting as an intensifier of "far".
There is a metaphor of "journeying" into the night, so "far on" means "a long way" (on the metaphorical journey into the night).
I have just come across this phrase while reading "How to Win Friends and Influence People" by Dale Carnegie.
"One man was so stirred by a talk on these principles that he sat far into the night discussing them with other members of the class. At three o’clock in the morning, the others went home. But he was so shaken by a realization of his own mistakes, so inspired by the vista of a new and richer world opening before him, that he was unable to sleep. He didn’t sleep that night or the next day or the next night – page 17
I think that "on" is redundant.
Below, you can find the definition provided by the Free Dictionary.
Far into the night
Extending until a late hour.