I have been in the software and internet world since the very beginning, I bought a Spectrum 8K when I was 13 (that was ages ago), I was already browsing the web when all content was for free - even without the need of using P2P - and I had never seen a company so successful, powerful, attractive and interesting like Google, of course, it is still not perfect.
Some, but not all, of the reason are described in this post.
Product superiority - Search engine is clearly superior compared the competitor's
Algorithm used to show search results is to do with relevance and popularity of the website and nothing else. Such algorithm, called PageRank, what comes from Larry's surname, one of the founders and owners of Google, has been the vehicle Google has used to grow so fast in the last ten years
Attract and retain talented people
It is the most valued company by technology professionals as a result of a sustained policy of motivating employees to innovate, research and feel part of the company. An example can be the 80-20 rule, meaning every employee can use 20% of their time in personal projects, things they like rather than things their boss likes. When one of that projects or ideas takes shape, it is reviewed by their colleagues and, eventually, by a steering committee where Schmidt (CEO), Page and Brin (founders) sit. Evaluation criteria for promoting an idea to become a funded project are never to do with profitability but with alignment with other products and with the global strategy of the company. One of the most "successful" Google services is news.google.com and no money is made out of it and, even though, it is consider a success. Most of the products they launched were conceived by people who had a need, one who needed a better mail tool, conceived Gmail, one who wanted to structure published information designed GNews and so on.
Don't be Evil
Google's strategy is mostly based in this principle "Don't be Evil" it is a very basic principle, far from the complex and not always clear mission, vision and values of some corporations. If Google had a mission, it is an idealistic one: "to make all of the world's information freely accessible and useful". For a Googler (Google's employee) "don't be Evil" means put the user first and put yourself in their shoes before writing a single line of code. If there is a conflict or doubt about the meaning of that principle, in words of Schmidt, "Evil is whatever Sergei says is evil" and Sergei sometimes is not so rational, as an example, type "wine" in google search and look at the right hand side of the search results page, you will see ads related to wine. Now, type "beer" in the search box and look at the ads area, you will see ... nothing, Sergei thinks beer is evil.
Google Guys stick in managing their company applying their criteria at any expense
When Google went public back in summer of 2004, they managed it on the way they wanted, at the expense of confronting the Wall Street establishment as broker firms profited with this IPO less than with others of same size. Unlike other technology companies, investors don't have decision power, they have, of course, influence but the trio Brin-Page-Schmidt (probably in that order) retain the decision making power.
Worldwide marketing and massive brand awareness for free
Word of mouth is the primary marketing tool Google has used, you think of how you discovered Google, was it because of any TV commercial? was it because of any website banner you clicked in? ... I bet it was because somebody told you how effective and easy to use it is.
Beside word of mouth, they use indirect marketing, what competitors and media say about them, about what they do and, most importantly, about how they do things, building around their brand a success and appealing halo. At the same time, they force all partners and collaborators to sign unusually strict non-disclosure agreement. Interesting enough was the digitisation of 50 million library books project, which will be completed by 2011 and where a few Universities were obliged to keep the project in secret until Google decide when and how it was going to be unveiled. That kind of secrecy was unusual in the educational environment and even put the project on risk but Google Guys always do things on their way, like Frank Sinatra.
In 2004 Google Mail was launched with great and unexpected (?) controversy on whether privacy rights of users were at risk. As you might be aware, but most of the users don't, there is an ads box (called sidebar link) on the right hand side of the screen when an email is opened. Ads are not selected randomly but, on the contrary, they are the result of a text scan operation and, then, a related "sponsored link" is shown to the user. This clearly means all your private information of your emails is known by the system. Google Guys listened to the arguments against such approach, consulted respected professionals and they decided not to move a letter or a line of code of the GMail product. That implied extending discussion and controversy about privacy what gave them more brand awareness and marketing for free rather than harmed them. We all know, even if you are not completely aware, that Google is not Evil, so they manage our privacy respectfully. They are the nice guys.