If science, robotics, and society progressed to the point where all human basic needs were provided for (food, clothing, shelter, health care, daily chores) at no cost and therefore nobody was required to labor, what would be of value?

Interesting! I may be misunderstanding the question, but you seem to suggest (or want to explore whether) labor is an essential measure or determinant of value, for the way you put matters is that if food, clothing, shelter, health care, daily chores, were not the result of labor, their value would be in question. I suggest, initially, that food, clothing, shelter, health care, daily chores, etc, are valuable whether or not they are the outcome of labor. It may be that we ourselves (psychologically) may not appreciate such values without them being tied into labor, but I propose that our lack of appreciation would then be the result of simply taking this satisfaction of basic needs for granted. As you did not specify the security or reliability of such non-labor satisfaction of basic goods, it might be added that our appreciation for their value may be quite vivid when we realize our vulnerability to theft, warfare, malicious destruction.

Another matter to consider: What kind of environment are you imagining? Is this a world in which persons have needs that go beyond the basic, material level, e.g. what about the need for individuals to be treated with respect and dignity? a world in which persons raise families? Fall in or out of love? Form political or social or religious parties? It may be that many of these good requires labor (or work or intentional striving). These might be goods that are good (have value) whether or not we labor or strive for them, but their realization may essentially require what could be called labor.