What work does God do daily?
First of all whatever work Supreme Lord does he has no purpose to do it from absolute level perspective which I discuss here and hence his activity is termed as Leela.
As for daily activity like what do Lord Shiva do daily in Kailash or what do Lord Vishnu do daily in Vaikuntha we can infer their daily activity from various Puranic sources and also from Itihasas. For eg. There are times when Lord Vishnu sleeps in the Milky Ocean. There are times when Lord Shiva is engaged in meditation.
Here is an example from Mahabharata about what Lord Shiva does:
He has snakes for his belt, and his ears are adorned with ear-rings made of snakes. Snakes form also the sacred thread he wears. An elephant skin forms his upper garment. He sometimes laughs and sometimes sings and sometimes dances most beautifully. Surrounded by innumerable spirits and ghosts, he sometimes plays on musical instruments. Diverse, again are the instruments upon which he plays, and sweet the sounds they yield. He sometimes wanders (over crematoria), sometimes yawns, sometimes cries, and sometimes causes others to cry. He sometimes assumes the guise of one that is mad, and sometimes of one that is intoxicated, and he sometimes utters words that are exceedingly sweet. Endued with appalling fierceness, he sometimes laughs loudly, frightening all creatures with his eyes. He sometimes sleeps and sometimes remains awake and sometimes yawns as he pleases. He sometimes recites sacred Mantras and sometimes becomes the deity of those Mantras which are recited. He sometimes performs penances and sometimes becomes the deity for whose adoration those penances are undergone. He sometimes makes gifts and sometimes receives those gifts; sometimes disposes himself in Yoga and sometimes becomes the object of the Yoga contemplation of others. He may be seen on the sacrificial platform or in the sacrificial stake; in the midst of the cow-pen or in the fire. He may not again be seen there. He may be seen as a boy or as an old man. He sports with the daughters and the spouses of the Rishis. His hair is long and stands erect. He is perfectly naked, for he has the horizon for his garments. He is endued with terrible eyes. He is fair, he is darkish, he is dark, he is pale, he is of the colour of smoke, and he is red. He is possessed of eyes that are large and terrible. He has empty space for his covering and he it is that covers all things. Who is there that can truly understand the limits of Mahadeva who is formless, who is one and indivisible, who conjures of illusions, who is of the cause of all actions and destructive operations in the universe, who assumes the form of Hiranyagarbha, and who is without beginning and without end, and who is without birth. He lives in the heart (of every creature). He is the prana, he is the mind, and he is Jiva (that is invested in the material case). He is the soul of Yoga, and it is that is called Yoga. He is the Yoga-contemplation into which Yogins enter. He is the Supreme Soul. Indeed Maheswara, the purity in essence, is capable of being comprehended not by the senses but through only the Soul seizing his existence. He plays on diverse musical instruments. He is a vocalist. He has a hundred thousand eyes, he has one mouth, he has two mouths, he has three mouths, and he has many mouths.
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