How old is Asta's son?

In the story, Asta's son is really Crispin, the illegitimate son of Lord Furnival.
The book begins in the year 1377. You'll find this date at the beginning of Chapter 1. In Chapter 3, Crispin (Asta's son) tells us that he was born in 1363 on the Feast of St. Giles. That would make Crispin 14 years old.
Crispin and his mother, Asta, lived in the village of Stromford until Asta's death. Before she died, Asta told Crispin that his father was killed during the Great Plague. The truth is that Crispin's father is really Lord Furnival, the lord of Stromford. However, Lord Furnival basely chooses not to acknowledge Crispin as his son.
As the story progresses, we learn that Lord Furnival is ailing; he eventually dies, causing those loyal to Lady Furnival to seek Crispin's death. For her part, Lady Furnival fears that Crispin will try to challenge her right to her husband's empire. This is why she sends John Aycliffe to hunt Crispin down. As for Crispin, he has little desire to make a claim on the Furnival wealth. He chooses instead to throw in his lot with Bear, an itinerant juggler who becomes a father figure to him.