An arbitrary and humorous alteration of million, billion, etc., the noun squillion denotes an extremely large but unspecified number, quantity or amount.

The earliest occurrences of the noun squillion that I have found are as follows, in chronological order:

1-: From a letter by Colonel Taggart, dated Portland, Oregon, Friday 16^th February 1872, published in the Public Press (Northumberland, Pennsylvania, USA) of Friday 20^th June 1879:

Before noon the next day, the 17^th, we were at Nanaimo taking in our supply of coal. Here I saw a sight that would have gladdened the heart of Agassiz. In every direction, as far as I could see around the ship, under the wharves, in all the little coves and inlets, the transparent waters were alive with fishes from four to six inches in length. One called them “smelts,” another said they were “perch.” At all events, within a radius of two hundred yards, were more fish of a single variety than there are human beings upon the face of the earth. It was almost a solid mass of life.
But for the occasional gleam of a white and glistening side, their close array might have been mistaken for the bottom of a very shallow water, not more than 18 inches deep, while in reality it was 30 feet. I was led to think, that in taking the census of Nature’s myriad forms of life, the inaccurate school-boy’s “squillions” would have to be mustered into the service. Talking of Nature’s myriad forms, did it ever occur to your agricolous imagination that several million “squillions” would scarcely enumerate the population of a single ten-acre field of midge-stricken wheat? and, if in the lapse of ages the descendants of these midges should be Darwinized into hyenas, tigers, wolves, alligators, boa-constrictors, vultures, Tammany politicians, turkey buzzards, and other voracious beasts and birds of prey, that they could not all make an honest living on this contracted sphere? At all events, obesity would be at a discount.

2-: From the Daily Illinois State Register (Springfield, Illinois, USA) of Wednesday 1^st September 1880:

For the Children.—Another lot of those $1.00 hammocks have been received at Robert’s, and are going off at the rate of a “squillion” a week.

3-: From the column Tea Table Gossip, published in The Evening News (Franklin, Pennsylvania, USA) of Friday 18^th January 1889:

—The statistician of the Meadville Tribune deserves a front place in the ranks of oil producers. In a comprehensive article he places the world’s consumption of oil at 75,000,000 bbls. a day. The product of Pennsylvania is put at 48,000,000 bbls. a day, being 27,000,000 bbls. a day less than the world’s demand. “The reserve stock above ground,” he says, “is estimated to be only sufficient for two years.” Basing a calculation on the foregoing figures, it would require at a low estimate a reserve stock of fully 9,420,000 bbls. to last for two years. This statistician’s figures are too modest. With a daily consumption of seventy-five million barrels, a larger reserve stock is needed. Why try to compute it in millions? It runs into squillions.

4-: From The Col. Sellers Club, published in the St. Louis Globe-Democrat (St. Louis, Missouri, USA) of Sunday 31^st March 1889:

“The fact of the matter is, gentlemen,” said a two-thirds owner in one of the most promising mining properties in the Western country, “the fact is that there is plenty of money in this world—bar’ls of it—millions on millions and squillions on squillions—but most of it is misplaced.”

The noun squillionaire designates a person who has an extremely large but unspecified amount of money.

The earliest occurrences of the noun squillionaire that I have found are as follows, in chronological order:

1-: From He Turns Out Songs: Willie Wildwave’s Factory Is a Queer Little Place, published in The Evening Post (Denver, Colorado, USA) of Wednesday 12^th June 1895:

The negro is more of a capitalist than his companion. He lays in 50 cents’ worth of stock and is a bit prone to jeer at his less opulent comrade.
“Look at de squillionaire!” he sneers with his hand on the door knob. “Yo’ mus’ hab spent ’s much as free cents all to once.”

2-: From The Genial Idiot, by the U.S. author John Kendrick Bangs (1862-1922), published in The Daily Northwestern (Oshkosh, Wisconsin, USA) of Saturday 23^rd April 1904:

“In a few years when it is known that the safe deposit boxes of this particular squillionaire are filled jam full of the manuscript of Moses, Homer, Milton, Shakespeare, Tennyson and Edwin Markham, that man will have a literary standing that will be apparently invincible.”