What activities can older people with limited mobility do that can keep them busy yet productive?

I feel it is not the activity but how you do it that will give you fulfillment. Fulfillment is a state of being, a state of realization. The desire to stay busy is because you don’t know how to remain still yet. The desire to be productive is related to the desire to feel accepted, to feel good enough. Both of these desires can be fulfilled by learning to find the source of acceptance within yourself, which will give you peace of mind also. With this love and peace as the basis of your being, you can engage in any activity your heart desires and which your limited mobility allows and you can be truly fulfilled, not because you are doing an activity, but by virtue of living with the source of fulfillment which is one with your self. The capability to experience joy by choice whether you are still or in activity can be learned. And if you discover this and live this, you can be an inspiration to other people to become happy as well. There is also an encompassing sense of responsibility associated with such an awakening, and one learns to connect to their inner intelligence to better be able to engage in whatever is most needed for this world, and choosing this path can make you more and more fruitful for the most amount of beings. Two sources I know that provide excellent education on this are Channel Higher Self and Isha Foundation. You can try New to Higher Self Teachings? Start here. page of the first site, or the Inner Engineering Program  of the second site to access excellent guides to most quickly start to educate yourself for discovering your inner abilities to create the fulfillment your heart desires to create.