This is the 499th most frequent Estonian word.
peaaegu
Almost.
In this sentence, 'peaaegu' describes how quickly the person recognized the answer, implying it was nearly immediate.
Ta teadis vastust peaaegu kohe, kui küsimust küsiti.
He knew the answer almost immediately after the question was asked.
Here, 'peaaegu' is used to indicate that the action of catching the bus was nearly achieved but ultimately not successful.
Ma jõudsin peaaegu bussi peale, kuid see sõitis minema.
I almost got on the bus, but it drove away.
Here, 'peaaegu' qualifies 'kõik', indicating that while most students were present, not all were.