All 18 Uses of

approach
in
Moby Dick

They stand in so great dread of some of them, that when out at sea they are afraid to mention even their names, and carry dung, lime-stone, juniper-wood, and some other articles of the same nature in their boats, in order to terrify and prevent their too near approach.
Chpt Extr

approach = getting near

However, hat and coat and overshoes were one by one removed, and hung up in a little space in an adjacent corner; when, arrayed in a decent suit, he quietly approached the pulpit.
Chpt 7-9 *

approached = moved toward

By some fishermen his approach is regarded as premonitory of the advance of the great sperm whale.
Chpt 31-33

approach = getting near

They contain round archipelagoes of romantic isles, even as the Polynesian waters do; in large part, are shored by two great contrasting nations, as the Atlantic is; they furnish long maritime approaches to our numerous territorial colonies from the East, dotted all round their banks; here and there are frowned upon by batteries, and by the goat-like craggy guns of lofty Mackinaw; they have heard the fleet thunderings of naval victories; at intervals, they yield their beaches to wild…
Chpt 52-54

approaches = comes toward

After taking counsel with his officers, he anchored the ship as far off shore as possible; loaded and ran out his two cannon from the bows; stacked his muskets on the poop; and warning the Islanders not to approach the ship at their peril, took one man with him, and setting the sail of his best whale-boat, steered straight before the wind for Tahiti, five hundred miles distant, to procure a reinforcement to his crew.
Chpt 52-54

approach = get near

Nor, though placed amongst a ruthless crew and every hour passed by ruthless hands, and through the livelong nights shrouded with thick darkness which might cover any pilfering approach, nevertheless every sunrise found the doubloon where the sunset left it last.
Chpt 97-99

approach = coming near

And not only this, but to that ever-contracting, dropping circle ashore, who, for any reason, possessed the privilege of a less banned approach to him; to that timid circle the above hinted casualty—remaining, as it did, moodily unaccounted for by Ahab—invested itself with terrors, not entirely underived from the land of spirits and of wails.
Chpt 106-108

approach = getting near

And here Bildad, who, with Peleg, be it known, in addition to his other officers, was one of the licensed pilots of the port—he being suspected to have got himself made a pilot in order to save the Nantucket pilot-fee to all the ships he was concerned in, for he never piloted any other craft—Bildad, I say, might now be seen actively engaged in looking over the bows for the approaching anchor, and at intervals singing what seemed a dismal stave of psalmody, to cheer the hands at the windlass, who roared forth some sort of a chorus about the girls in Booble Alley, with hearty good will.†
Chpt 22-24

In the length he attains, and in his baleen, the Fin-back resembles the right whale, but is of a less portly girth, and a lighter colour, approaching to olive.†
Chpt 31-33

And it is much to be deplored that the place to which you devote so considerable a portion of the whole term of your natural life, should be so sadly destitute of anything approaching to a cosy inhabitiveness, or adapted to breed a comfortable localness of feeling, such as pertains to a bed, a hammock, a hearse, a sentry box, a pulpit, a coach, or any other of those small and snug contrivances in which men temporarily isolate themselves.†
Chpt 34-36

I know that, to the common apprehension, this phenomenon of whiteness is not confessed to be the prime agent in exaggerating the terror of objects otherwise terrible; nor to the unimaginative mind is there aught of terror in those appearances whose awfulness to another mind almost solely consists in this one phenomenon, especially when exhibited under any form at all approaching to muteness or universality.†
Chpt 40-42

But not so did it seem to Ahab, who knew the sets of all tides and currents; and thereby calculating the driftings of the sperm whale's food; and, also, calling to mind the regular, ascertained seasons for hunting him in particular latitudes; could arrive at reasonable surmises, almost approaching to certainties, concerning the timeliest day to be upon this or that ground in search of his prey.†
Chpt 43-45

"There is some sneaking Temperance Society movement about this business," he suddenly added, now approaching Starbuck, who had just come from forward.†
Chpt 70-72

In vain handspikes and crows were brought to bear upon the immovable fluke-chains, to pry them adrift from the timberheads; and so low had the whale now settled that the submerged ends could not be at all approached, while every moment whole tons of ponderosity seemed added to the sinking bulk, and the ship seemed on the point of going over.†
Chpt 79-81

As marching armies approaching an unfriendly defile in the mountains, accelerate their march, all eagerness to place that perilous passage in their rear, and once more expand in comparative security upon the plain; even so did this vast fleet of whales now seem hurrying forward through the straits; gradually contracting the wings of their semicircle, and swimming on, in one solid, but still crescentic centre.†
Chpt 85-87

The lost life-buoy was now to be replaced; Starbuck was directed to see to it; but as no cask of sufficient lightness could be found, and as in the feverish eagerness of what seemed the approaching crisis of the voyage, all hands were impatient of any toil but what was directly connected with its final end, whatever that might prove to be; therefore, they were going to leave the ship's stern unprovided with a buoy, when by certain strange signs and inuendoes Queequeg hinted a hint concerning his coffin.†
Chpt 124-126

So Ahab's proceedings in this matter were not unusual; the only strange thing about them seemed to be, that Starbuck, almost the one only man who had ever ventured to oppose him with anything in the slightest degree approaching to decision—one of those too, whose faithfulness on the look-out he had seemed to doubt somewhat;—it was strange, that this was the very man he should select for his watchman; freely giving his whole life into such an otherwise distrusted person's hands.†
Chpt 130-132

Definitions:

(1) (approach as in: approached the city)
to get closer to (near in space, time, quantity, or quality)
(2) (approach as in: use the best approach)
a way of doing something; or a route that leads to a particular place
(3) (approach as in: approached her with the proposal)
to begin communication with someone about something -- often a proposal or a delicate topic
(4) (meaning too rare to warrant focus)
meaning too rare to warrant focus:

More rarely (and typically only in classic literature), the phrase nearest approach to as used in "her nearest approach to an apology" or "her nearest approach to a smile" typically means that "something is as close to something else as it ever gets." "As near an approach to" can have a similar meaning.