People are usually surprised when they learn that the longest soliloquy in Shakespeare doesn't occur in Hamlet, Othello or any of the other plays associated with the technique - I know I was when I discovered that it occurs in the second scene of the third act of Henry VI, Part Three where it is spoken by the third son of the Duke of York, Richard, the recently nominated Duke of Gloucester. In it, the audience discovers that the character, like his father before him (mocked and killed by Lancastrian nobles in a brutal scene in 1.4) has plans to take the throne, but in this case, it's his older brothers more than the House of Lancaster that are in his way. The soliloquy contains many of the qualities that we associate with later ones: the working out of problems and decisions in front of the audience, the psychological realism, as well as the sources of what would become known as the stream-of-consciousness technique. And it's interesting to note that the three Henry VI plays are panoramic in effect, with many important roles, but with none bigger than 400 words or so. But this speech was a turning point for Shakespeare, and this character (who becomes better known as Richard III, and who speaks 1171 lines in the play named for him that follows) was the impetus for it. Here's an excerpt from the soliloquy in question:

Why, then, I do but dream on sovereignty;

Like one that stands upon a promontory,

And spies a far-off shore where he would tread,

Wishing his foot were equal with his eye,

And chides the sea that sunders him from thence,

Saying, he'll lade it dry to have his way:

So do I wish the crown, being so far off;

And so I chide the means that keeps me from it;

And so I say, I'll cut the causes off,

Flattering me with impossibilities.

My eye's too quick, my heart o'erweens too much,

Unless my hand and strength could equal them.