Realize that large corporations came to being by starting as small companies, and by carefully growing their user base. Often large corporations with existing product lines have been around for a while, so they have a number of institutional shortcomings which can be exploited for competitive purposes.
This is especially true in the arena of software, where progress over the past 10 years have left older companies with legacy solutions for problems that no longer exist.
Also realize that most companies have, in an effort to fill their niche and exploit as many revenue streams as possible, have expanded their products by adding features that very few people use. Often products like this (such as Microsoft Office) are attempting to tick off as many checkmarks on the product feature list as possible in order to justify their high price. From a cost benefit perspective, however, those additional features cost a tremendous amount to develop, in order to capture an increasingly smaller audience–so the companies are in effect using the existing revenue stream to develop features that add very little value to capture a fractional number of people.

So to slay a dragon, or at least to compete against one:
(1) Look for products where the established player has an expensive and complex product, with complexity that was created to solve problems (such as memory limitations or computational limitations) which no longer exist. For example, Photoshop was designed in an era where 8 megabytes of RAM was unheard of, and where processor speeds were measured in dozens of megahertz. So Photoshop is full of legacy image paging and image management utilities that are just no longer needed in an era where the cheapest throwaway laptops have 100 times more RAM and run 100 times faster.
(2) Look for products which have a tremendous amount of complexity, where that complexity has been in part replaced by advancements in operating system design or which is just no longer needed because of advances in hardware. Again, looking at Photoshop, much of Photoshop’s image processing abilities have been duplicated by MacOS’s CoreGraphics engine, which means what once required a Ph.D. in Computer Graphics to code can now be handled by any noob with a few calls to a well-documented API. (The underlying algorithms are still complicated, but they’re delivered to you on a silver platter.)
(3) Look for products where 90% of the users only use 10% of the features.
When these three things are true, you have a perfect storm: the ability to create a product (like Pixelmator) which is inexpensive (because you don’t have to recreate the entire Photoshop computational stack), which does 10% of what Photoshop does (but the 10% that people want), which can be sold for a fraction of what Photoshop sells for.
The brothers who created that tool made millions.
Of course it takes decades to overthrow an Adobe or a Microsoft Word. But it is doable: we’ve already seen it with Quark XPress and with Framemaker, and I believe we’re seeing it right now with Microsoft Office (at least in the Macintosh ecosystem) being displaced by iWork.
Sometimes you can even tackle a market by creating a product that does 10% of a competitor’s functionality, but adds a few new features a competitor would never add in a million years, to attack a new market no-one is really addressing. For example, Opacity is a brilliant little tool that sorta competes against Illustrator, but is really designed for building the little fiddly bits of artwork used for application development. (To tell you how application-centric that tool is, it even has an export utility which will export Objective C code to draw the paths, and has the ability to insert variables that get expressed in code, so you can draw artwork that then is translated into a function call for drawing a UIView of an NSView.)

It is possible to slay dragons. You just have to be careful how you do it. It takes a long time. It’s risky.
But just because some big guy is in that market already doesn’t mean you can’t take that market over and eventually win. After all, Apple was some stupid PC maker who didn’t know anything about the consumer market or how cell phones are made.