Our world has easily observed year, month and day cycles. They and their multiples, such as decades or weeks, and fractions, such as seasons, hours, minutes, and seconds are easily available and used to synchronize and measure events. Seasons and daylight make timing actions highly important activity as well since the earliest times. Most animals and even plants actually have biological clocks.
But few recent questions have been about worlds without such easily observed cycles. This includes constant daylight and no seasonal variations.
Most answers focused on alternate ways of measuring time or synchronizing actions. But I got to thinking...
What if instead society responded to the lack of external events allowing and requiring synchronization by simply going asynchronous. With computers we use, depending on need and circumstance, both synchronous and asynchronous protocols. Our society also uses both, but synchronous approaches tend to dominate especially in modern times when cheap and accurate clocks are available. Less modernized cultures were more laissez-faire about exact schedules and work hours.
A society without days or seasons would probably go the opposite way than we have. Since measuring and determining time would be more difficult and less critical, they'd probably avoid it as much as possible.
But how much of it is possible to avoid? Can a society be wholly asynchronous and avoid entirely the need to measure time? If no, why not? If yes, how would such society work? Would the answer vary based on level of technology?
You can assume full biological adaptation by people, animals, and plants.
type_outcast pointed that heart beats would still allow and supply a measure of time at short time scales. This would naturally extend to things such as music and poetry having beat and tempo, and the length of recitation or song providing a way to estimate lengths of time. Further this would then naturally be extended to using pendulums and mechanical clocks.
I don't really see any reason why the people would not do this or why I or anyone else should assume they won't. What the question is about is the social protocols people would use without the larger structure of years and days and without a convenient source of assumed synchronicity. There is no need to assume they would be unable to measure periods of time.
This would imply that after mechanical clocks become accurate they would have ability to synchronize things if they chose to. But let's just ignore that since it is the solutions they'd have developed before that that are interesting. And in any case they'd still not organize around days or years, so I'd assume the old ways would still be better.
EDIT 2: Adding a link to comment by slebetman in one of the answers about just-in-time processes and Toyota way. I should have mentioned all that in the question from the beginning, but it didn't occur to me. However it is a very good illustration of the kind of use of asynchronous protocols I am talking about. (Ignore the fact that in our world synchronous parts are necessary at the interfaces.)