Old English great "big, tall, thick, stout, massive; coarse," from West Germanic *grauta- "coarse, thick" (source also of Old Saxon grot, Old Frisian grat, Dutch groot, German groß "great"). If the original sense was "coarse," it is perhaps from PIE root *ghreu- "to rub, grind," via the notion of "coarse grain," then "coarse," then "great;" but "the connextion is not free from difficulty" [OED].
It took over much of the sense of Middle English mickle, and itself now is largely superseded by big and large except in reference to non-material things. In the sense of "excellent, wonderful" great is attested from 1848.
Great White Way "Broadway in New York City" is from 1901, in reference to brilliant street illumination. The Great Lakes of North America so called by 1726, perhaps 1690s. Great Spirit "high deity of the North American Indians," 1703, originally translates Ojibwa kitchi manitou. The Great War originally (1887) referred to the Napoleonic Wars, later (1914) to what we now call World War I (see world).
"The Great War" — as, until the fall of France, the British continued to call the First World War in order to avoid admitting to themselves that they were now again engaged in a war of the same magnitude. [Arnold Toynbee, "Experiences," 1969]
Also formerly with a verb form, Old English greatian "to become enlarged," Middle English greaten "to become larger, increase, grow; become visibly pregnant," which became archaic after 17c.
"sharpshooter; one who shoots from a hidden place," 1824, agent noun from snipe (v.). The birds were considered a challenging target for an expert shooter:
Snipe Shooting is a good trial of the gunner's skill, who often engages in this diversion, without the assistance of a dog of any kind; a steady pointer, however, is a good companion. [Sportsman's Calendar, London, December 1818]