This is the 538th most frequent Hungarian word.
végére
"Végére" means "to the end" in English.
In this sentence, 'végére' is used metaphorically to indicate the final part of a process or timeframe.
A dolgozat vége felé elkezdett csökkenni a türelme.
Towards the end of the essay, his patience started to decrease.
In this sentence, 'végére' is used to indicate arriving at the conclusion or endpoint of something, in this case, a story.
Elérkeztünk a történet végére.
We have reached the end of the story.
Here, 'végére' specifies the end point or destination of the journey described.