Early mystic experiences of Sri Aurobindo
In the formative years of our youth, some of us experience those serendipitous and decisive moments which open new vistas and reveal to us our calling in life; one individual may chance upon a stirring piece of music and be impelled to become a musician; another may be captivated by an enigmatic pattern of numbers and subsequently enroll in math studies; and yet another might discover an uncanny aptitude for mechanical tools and go on to build a business around it. These are predestined moments, moments when the soul shines forth to disclose our purpose and guide us to our vocation in this incarnation. In the lives of mystics as well, we see such transcendental moments which initiate their entry into the spiritual path. This article presents some early mystic experiences of Sri Aurobindo as noted down in his poems.
Before we discuss Sri Aurobindo, let me digress by providing an example from the early life of a noted computer scientist Michael O. Rabin. In this excerpt from the book Out of their minds: the lives and discoveries of 15 great computer scientists by Dennis Shasha, Michael Rabin narrates an episode in his childhood which motivated him to study math even though his father wanted him to devote himself to religious studies.
My sister, who is five years older than I, brought home The Microbe Hunters by Paul de Krif. Reading that and other books on the pioneers of microbiology sparked my imagination, so from the time I was eight until about twelve, I thought I might become a microbiologist.
Then one day serendipity played a role and I was kicked out of class. There were two ninth-grade students sitting the corridor solving Euclid style geometry problems. I looked at what they were doing. There was a problem they couldn’t solve. So they challenged me and I solved it. The beauty of that, the fact that by pure thought you can establish a truth about lines and circles by the process of proof, struck me and captivated me completely.  (link)
Such transformative experiences abound in the lives of mystics as well. Ramakrishna Paramahansa early in his childhood was said to have been entranced by the ethereal sight of dazzling white geese silhouetted against a dark monsoon sky. Rabindranath Tagore in his youth had a vision of the world bathed in light; this vision informed his soul-stirring poetry and philosophy of life.
Early experiences of Sri Aurobindo
Sri Aurobindo was born in 1872 in Calcutta, India. At the tender age of seven, he was sent to England by his father for higher studies and that is where he remained till the age of twenty-one. While in England, he once came across a brief statement on the “Six philosophies of India” and was especially struck by the concept of the Atman in Advaita. He thought it might bear a clue to the reality behind the phenomenal world but he couldn’t ascertain its deeper significance at the time. 
He returned back from England to India in February of 1893 on board a steamship as was the custom in those days. When the ship docked at Apollo Bunder (seen below) in Bombay(now Mumbai), he felt a vast calm descending upon him. This envelope of calm that ensconced him persisted for many months. 
He subsequently took up employment with the Maharaja of Baroda State. In the first year of his stay(approx 1893-1894), he was involved in a horse carriage accident during which he had the vision of a Godhead surging up from within to save his life.  He wrote the poem “The Godhead” describing this experience.
I sat behind the dance of Danger’s hooves
In the shouting street that seemed a futurist’s whim,
And suddenly felt, exceeding Nature’s grooves,
In me, enveloping me the body of Him.
Above my head a mighty head was seen,
A face with the calm of immortality
And an omnipotent gaze that held the scene
In the vast circle of its sovereignty.
His hair was mingled with the sun and breeze;
The world was in His heart and He was I:
I housed in me the Everlasting’s peace,
The strength of One whose substance cannot die.
The moment passed and all was as before;
Only that deathless memory I bore.