I'd say that you should have a combination of experience with a stack of technologies and be prepared to experiment some with new things once in a while. This falls under the heading of "continuous improvement" or "professional development" where you may have an idea that you want to try out, e.g. is using LINQ or generics really better than the older style for things.
Myself, I come from the land of Microsoft tools. I have used every Visual Studio from 5.0 to 2008, have programmed in C/C++, VBScript, C#.Net, JavaScript and used mark-up languages like XML, HTML and propietary ones in some places. I've also used MS-SQL from 6.5 to 2005 and am comfortable creating tables, stored procedures and various other database tasks like creating a view. Does this mean I'll never touch Java? Hardly, but when I do it will likely be because something around me is starting to use it and I have to learn part of it, to do my job well.
When it came time to learn ASP and ASP.Net, I had employers that were willing to give us a break in that we'd be self-taught in these technologies as sometimes if the technology is new enough, it isn't like there will be courses in it within the first few months. For example, I remember one place developing with a beta version of Visual Studio 2005 for their main enterprise application. It was rather cool and for some things like researching AJAX frameworks this became moot, but I did get the experience of getting out there.
I've also had employers where there is that downtime of I can't write the code for features ABC until the back-end guys get the API together and so I go on to other things and generally in the background keep an eye on some things and where necessary, invest my own time and effort into getting my hands on various technologies through local user groups and Microsoft events.
Another example here is that say someone is reporting a problem with application ABC that was built in-house. In investigating the problem, you see that the code should be refactored and think it will take a few weeks, one for the fix itself but a couple more to get the code up to a point where it is easily maintainable. In suggesting a fix, say you'll need three weeks and then do what you said you'd do. Having integrity and doing what you say and meaning what you say is an important soft skill that I think gets lost by some folks at times.
I would say that my field has evolved and will likely continue to do so, forever. When I started after university, I was programming in ISAPI extensions and learning IIS as this is what the dot com where I was had for their systems. I also picked up MS-SQL Server while I was there as something else to get my hands on and through the turnover of many engineers, I moved around a little into a web team when there was one and used ASP and eventually ASP.Net before it went under totally. More than a few of the technologies I picked up there I still use like Visual Studio and SourceSafe to name another couple of things I got there. After a couple of dot-coms, I shifted into an Application Service Provider that was a different field than the Internet retailers I had before. Now, I'm working for a technology company and while the processes have changed from place to place and various tools have changed, some things still remain the same: The customers/clients/business users want features A,B and C in some application that is either home-grown or customized to the company and so we go in and tweak it for them to use. That part of the job hasn't changed and I doubt it will ever go away.
Edit:
Just to add another part to this in terms of getting jobs, here are a few ideas:
Get to know some recruiting firms when you don't really need them. See who wants to get to know what you really want in a job and who you are, etc. You never know when that may be helpful. These can give you a way around HR as the recruiters are talking to hiring managers directly.
Understand that your ability to work with various companies will likely change over time, unless you start your own. Even companies like Microsoft and Google have had layoffs and what was once a great company may lose that status as every company has its painful years,e.g. look at Apple in the late 1990s. If you create your own company you will still have to find clients or customers to pay the bills.
Know what you can do well and what you like to do. While some HR folks may not like this, there are some places where they want to see your passion or enthusiasm in order to take that leap of faith of hiring you.
Maintain a network or two. Sites like LinkedIn can be useful for this, just to see who is up to what and possibly to ask if anyone is looking for someone who can do ABC. This goes along with the recruiter point as really, those great jobs aren't always going to come to you.