Word to describe an absorbing activity which causes one to fail to notice the passage of large amounts of time

Engrossing, perhaps.

engrossing adjective

very interesting and needing all your attention:

• an engrossing book/story
• I found the movie completely engrossing from beginning to end.

Cambridge Dictionary

The activity allows one to enter flow, "the mental state of operation in which a person performing an activity is fully immersed in a feeling of energized focus, full involvement, and enjoyment in the process of the activity" (link and quote from Wikipedia).

For many, including myself, when I enter a period of flow, it is only ended by the sudden and surprising realization that I have not eaten, urinated, or performed other basic needs for that entire time, and out of nowhere they have become urgent.

Some games, a challenging puzzle, some tasks at work (I write software code) and such have caused me to experience this type of flow.

"engross" is the best word for it, as suggested.

You could also use the expression "to lose oneself in (something)".