What is the "scientific" purpose of existence?

First, let us establish that existence requires the involvement of a living being, by definition, not uncontroversially. Now we can look specifically to the purpose of life, not just existence.

Scientifically, and if you'll excuse the tautologies, life's purpose is to live: the objective of a living being is to be, a state maintained with unconscious intentionality and care (Sorge). Even at the most basic level, as soon as a basic life form is about to stabilise, its main aim (so to speak) is to stay alive, or it doesn't qualify.

On a loarger scope Nietzsche referred to this as the will to power. Freud wrote about the Life drive and the Death drive (Eros & Thanatos) in Beyond the Pleasure Principal. Derrida combine all three as a Life drive in "To Speculate - On Freud" in The Post Card. (The death drive (repetition compulsion) is seen as a flip side of unconscious, tenacious problem solving on the one hand, and belligerent tribal rutting instincts on the other, so death drive is seen as toughening, and paradoxically also a life drive.)

A rational being knows it will one day die, so its refined aim would be to live optimally within its indeterminate time.

Science is not well adapted to addressing purpose -- most science starts with the assumption that purpose should be either ignored, or dismissed. So --what you would be interested in may be scientists who are bringing their skill set to philosophy. This will not be a single reference, as these are diverse thinkers, with diverse subjects they address.

I can offer several suggestions. The first is not a scientist, but it most directly addresses intentionality. It is Thomas Nagel's Mind & Cosmos. I have a review here: https://www.amazon.com/gp/customer-reviews/R1HQMF6OJKRDCC?ref=pf_ov_at_pdctrvw_srp

Another is Freeman Dyson's Infinite in all Directions. This is Freeman exploring a tossed salad of speculative ideas.

Yet another is Unger and Smolin's The Singular Universe. It does not touch on intentionality or consciousness, but does offer an approach to thinking about math and logic as real AND dependent on matter. I have a review here: https://www.amazon.com/gp/customer-reviews/R16VWWZ5I5SC8Q?ref=pf_ov_at_pdctrvw_srp

And a fourth would be Popper and Eccles The Self and Its Brain. Consciousness, reality, and how to fit both into an ontology, with a Nobel prize winning scientist collaborating with the premier philosopher of science.

Another -- Victor Stenger's Has Science Found God? He says no. Stenger discusses entropy, and how an expanding universe allowed life to develop despite entropy. I review here: https://www.amazon.com/gp/customer-reviews/RIXZJM4IVJ8AN?ref=pf_ov_at_pdctrvw_srp

Yet another -- Paul Davies The Mind of God, reviewed here, looks at a collection of questions: https://www.amazon.com/gp/customer-reviews/R14GY1JWNITAOJ?ref=pf_ov_at_pdctrvw_srp

And -- a diverse team of scientists try to tackle consciousness and ontology, in Beyond Physicalism, which I review here: https://www.amazon.com/gp/customer-reviews/RZY1A4EL2JOZ4?ref=pf_ov_at_pdctrvw_srp

Also -- here is a quick summary of "scientific" (IE Darwinian) approaches to morality: https://www.meetup.com/Philosophy-Cafe-Central-Maryland/events/qhvxpqybcgbcb/