Shakespeare knew the different definitions of the word "game", and used them to refer to actual games as well as games of politics, courtship, or intrigue. He also often used "gamester" to refer to a gambler or a chance-taker.
The first two parts of As You Like It's first act revolve around a wrestling match between Orlando and Charles.
In Troilus and Cressida, Nestor makes a brief offhand mention of the sport: "And I have seen thee pause and take thy breath, When that a ring of Greeks have hemm'd thee in, Like an Olympian wrestling: this have I seen; But this thy countenance, still lock'd in steel, I never saw till now."
Another offhand reference to Olympian games is given in Henry VI, Part 3 by George: "And, if we thrive, promise them such rewards As victors wear at the Olympian games: This may plant courage in their quailing breasts; For yet is hope of life and victory."
In Measure for Measure, "tick-tack" literally means the game Tic Tac Toe, although euphemistically it refers to sexual play. Lucio worries that Claudio's sister (Isabella) should be kept in a chaste environment: "I pray she may; as well for the encouragement of the like, which else would stand under grievous imposition, as for the enjoying of thy life, who I would be sorry should be thus foolishly lost at a game of tick-tack."
Dice games were popular at the time. Not only are dice mentioned by name, but specific dice games, such as Hazard, also weigh the meaning of words on occasion (source).
In the Merry Wives of Windsor, Shallow, a Country Justice refers to dice: "How now, master Parson! Good morrow, good Sir Hugh. Keep a gamester from the dice, and a good student from his book, and it is wonderful."
In the Merchant of Venice, Morocco, one of Portia's suitors, reflects on his chances: "If Hercules and Lichas play at dice Which is the better man, the greater throw May turn by fortune from the weaker hand."
In Much Ado About Nothing, the feisty Beatrice reflects on losing Benedick's heart: "Indeed, my lord, he lent it me awhile; and I gave him use for it, a double heart for his single one: marry, once before he won it of me with false dice, therefore your grace may well say I have lost it."
Falstaff recounts his wild days in Henry IV, Part 1: "I was as virtuously given as a gentleman need to be; virtuous enough; swore little; diced not above seven times a week; went to a bawdy-house once in a quarter--of an hour; paid money that I borrowed, three of four times; lived well and in good compass: and now I live out of all order, out of all compass."
King Lear's Edgar does the same: "A serving-man, proud in heart and mind; that curled my hair; wore gloves in my cap; served the lust of my mistress' heart, and did the act of darkness with her; swore as many oaths as I spake words, and broke them in the sweet face of heaven: one that slept in the contriving of lust, and waked to do it: wine loved I deeply, dice dearly: and in woman out-paramoured the Turk: false of heart, light of ear, bloody of hand; hog in sloth, fox in stealth, wolf in greediness, dog in madness, lion in prey."
In Henry V, the chorus describes the English and French attitudes: "Proud of their numbers and secure in soul, The confident and over-lusty French Do the low-rated English play at dice; And chide the cripple tardy-gaited night Who, like a foul and ugly witch, doth limp So tediously away.
Later, Dauphin decries his side's broken ranks: "O perdurable shame! let's stab ourselves. Be these the wretches that we play'd at dice for?"
Biron, one of the three men swearing chastity in pursuit of scholarship in Love's Labours Lost, refers to dice twice. The first is while bantering with Princess: "Nay then, two treys, and if you grow so nice, Metheglin, wort, and malmsey: well run, dice!"
And again while describing his friend Boyet: "This gallant pins the wenches on his sleeve; Had he been Adam, he had tempted Eve; A' can carve too, and lisp: why, this is he That kiss'd his hand away in courtesy; This is the ape of form, monsieur the nice, That, when he plays at tables, chides the dice In honourable terms: nay, he can sing A mean most meanly; and in ushering Mend him who can: the ladies call him sweet;..."
Even Hamlet has a passing reference to dicers: "Such an act That blurs the grace and blush of modesty, Calls virtue hypocrite, takes off the rose From the fair forehead of an innocent love And sets a blister there, makes marriage-vows As false as dicers' oaths ..."
In Anthony and Cleopatra, a Soothsayer gives advice to Mark Anthony about Caesar: " If thou dost play with him at any game, Thou art sure to lose; and, of that natural luck, He beats thee 'gainst the odds: thy lustre thickens, When he shines by: I say again, thy spirit Is all afraid to govern thee near him; But, he away, 'tis noble."
Mark Anthony reacts: "Be it art or hap, He hath spoken true: the very dice obey him; And in our sports my better cunning faints Under his chance: if we draw lots, he speeds; His cocks do win the battle still of mine, When it is all to nought; and his quails ever Beat mine, inhoop'd, at odds."
Mark Anthony later decries Cleopatra's alliance with Caesar: "I made these wars for Egypt: and the queen,-- Whose heart I thought I had, for she had mine; Which whilst it was mine had annex'd unto't A million more, now lost,--she, Eros, has Pack'd cards with Caesar, and false-play'd my glory Unto an enemy's triumph."
In King John, Louis of France will not make peace with England: "Have I not here the best cards for the game, To win this easy match play'd for a crown?"
Some information on Elizabethan games. The page looks like a spam site; maybe it is.
Gambling in Shakespeare's time. Card games. And more games. Other sources abound.