This is the 1036th most frequent Catalan word.
quasi
Almost.
In this sentence, 'quasi' modifies 'tots' to express that the majority but not absolutely all arrived on time.
Quasi tots els assistents van arribar puntuals.
Almost all the attendees arrived on time.
Here, 'quasi' is used to convey a sense of hardly or just barely in the described situation.
Era tan impressionant que quasi no vaig poder parlar.
It was so impressive that I could barely speak.
The word 'quasi' here indicates that the action of finishing is nearly complete.