Is there a word that means near-daily?
Solution 1:
The phrase most days is widely used: "Most days we have a staff meeting."
Solution 2:
I don't know of a word that means "near-daily" or "most days". Besides those terms, consider "almost-daily", "at most daily", and "daily (as needed)". If the task is always performed at the same time of day, you might refer to "the X task (as needed)" where X is, for example, dawn, morning, noon, afternoon, evening, or a specific time.
Usually and related words lead to phrasings such as nominally daily, normally daily, notionally daily, ordinarily daily, and usually daily.