As many as you need to get the job done (and then some). If programming is the right career choice for you, you will be thrilled to learn new languages and platforms. A few programmers work for many years on the same platform using the same language. But most of us have to frequently learn new languages; often with each new system to be developed. It is a never-ending race to keep up with the latest and best technologies. To further add to the challenge, what was popular and highly recommended just a few years ago is now considered archaic or even not recommended for most usage (e.g.: applets, EJBs, web applications with multiple frames, VB). You will have to get used to this constantly changing, ever-evolving state of affairs in programming--it will only intensify as time goes by. If you are unsure what you should know, look at the languages that have been in use, and are still popular, after 10 years: C/C++, Java, SQL, Perl, etc....
If it's any comfort for you to know: with each year of programming, you will find that learning a new language, even one with a very different syntax or paradigm than what you are accustomed to working with, will become easier and easier. The same themes and ideas will keep popping up. What most creators of new languages are trying to do is find ways to make it easier to program systems in a way that is more natural in terms of logical thought, in fewer lines of code.