In his famous monologue (As You Like It, Act II Scene vii), Jacques doesn't actually use alliteration as a primary rhetorical device. In fact, I think it might be more beneficial to ask why Jaques doesn't use it, as it is a common and easy way to create powerful wordplay—one that Shakespeare takes great advantage of elsewhere.
Just to make sure we're on the same page: alliteration is the repetition of the same sound at the start of closely connected words. The phrase I just used, "same sound at the start," is a perfect example of alliteration—the "s" sound starts three of the five words.
Alliteration is important because our brains are programmed to pay special attention to patterns. When we hear repetition, we grab onto it. Of course, there are lots of different things authors can repeat—vowel sounds (that's called assonance), consonant sounds (consonance), end sounds (rhymes), entire words—but repeating the starting sound of words is especially effective because that's the first thing we hear. We're zeroed in on the phrase from the beginning, and therefore likely to catch its full meaning (or meanings).
Repetition creates links between words. In the early days of English storytelling—we're talking Beowulf here—repetition was useful because the stories were delivered orally, from memory, rather than written down, and alliterative phrases were easy to remember. By the time Shakespeare rolled around, authors were putting a lot more effort into crafting beautiful-sounding passages; alliteration still helped the actors with memorization, but it also functioned as ornamentation and signal: this bit is important.
In Jaques' speech, there are two big alliterative moments. Perhaps not coincidentally, they are the two most recognizable parts of the monologue: the first four lines and the last one.
He starts this way. I'll highlight the alliteration in bold.
All the world's a stage,
And all the men and women merely players;
They have their exits and their entrances,
And one man in his time plays many parts
The first two lines feature a lot of As. Even so, it's hard to say that Jaques is consciously alliterating. He repeats "all," but the concept itself is the important takeaway, not the letter—he's underscoring that this speech will encompass everything about life. The other A words seem incidental—"a" and "and."
The Es and Ps are linking concepts. Exits and entrances are antitheses—opposites (another of Shakespeare's favorite devices). Parts are what players play. The alliteration is relevant, but it's gone too quickly to feel significant.
Here's Jaques' last line. He's referring to the seventh "part" a man plays in his life, old age, which ends in a state that leaves the man ...
Sans teeth, sans eyes, sans taste, sans everything.
This is the most relevant piece of alliteration in the monologue because it's serving a clever double purpose. The repetition of the word "sans" (Latin for "without") emphasizes that one dies completely alone and stripped of everything—worldly goods, friends, a functioning body. But think about how you'd talk without teeth. One of the few sounds you'd be able to make is a hiss. The repeated Ss (notice that "sans" also uses consonance) are onomatopoeia. Jaques is mimicking the sound of a toothless old man trying to talk, even as he's describing the indignities he suffers.
Elsewhere in the monologue, there are a few isolated instances of alliteration, but they occur too rarely to be a trend. So why, in a 28-line speech, is that all we get?
I'd suggest we look in two directions. First, consider the subject of the monologue. Jaques is describing the seven stages of life every man goes through: infant, school-boy, lover, soldier, justice, pantaloon, old man. (Note that he only refers to men, not women, and that, though he says one man plays "many parts," he goes on to imply that every man plays seven, no more, no less, and always in the same order.)
He's saying that life proceeds linearly from point to point. Things don't repeat; they constantly evolve. If Jaques used repeated words or sounds, then, it would undermine the central point of his argument. (The exception is old age, which he calls a "second childishness," and which features, as we mentioned, the only significant alliteration in the speech.)
Second, consider what Jaques is doing. He bounds into II.vii. unusually happy. When Duke Senior asks why, Jaques explains that he met Touchstone, a court jester. Jaques was thrilled to hear Touchstone's wordplay, and decided that he wanted to become a fool himself. Listen to this speech of his, which comes earlier in II.vii. than the Seven Ages of Man monologue. Notice how often Jaques repeats words and uses alliteration.
A fool, a fool! I met a fool i' the forest,
A motley fool; a miserable world!
As I do live by food, I met a fool
Who laid him down and bask'd him in the sun,
And rail'd on Lady Fortune in good terms,
In good set terms and yet a motley fool.
'Good morrow, fool,' quoth I. 'No, sir,' quoth he,
'Call me not fool till heaven hath sent me fortune:'
And then he drew a dial from his poke,
And, looking on it with lack-lustre eye,
Says very wisely, 'It is ten o'clock:
Thus we may see,' quoth he, 'how the world wags:
'Tis but an hour ago since it was nine,
And after one hour more 'twill be eleven;
And so, from hour to hour, we ripe and ripe,
And then, from hour to hour, we rot and rot;
And thereby hangs a tale.' When I did hear
The motley fool thus moral on the time,
My lungs began to crow like chanticleer,
That fools should be so deep-contemplative,
And I did laugh sans intermission
An hour by his dial. O noble fool!
A worthy fool! Motley's the only wear.
Remember that Shakespeare's fools are often among the most perceptive characters in the play, couching their observations in witty or nonsensical sayings. They're also some of the most upbeat, appearing relentlessly cheerful and energetic, allowing insults to bounce off them.
Jaques, on the other hand, is notoriously mopey and broody, at times even misanthropic. Definitely not natural-born fool material. When he quotes a true fool, he manages to flash some skill with words. But when he tries to come up with his own material—the Seven Ages of Man speech, for example—he can't muster the same verve.
It's ironic, but one of Shakespeare's most beloved passages might actually be intentionally mediocre. Perhaps there isn't more alliteration simply because the speaker doesn't have the rhetorical chops to deliver A-grade material.