To simulate human experience and thought sufficiently well that we may find a way to survive the inability for our bodies to survive an ecological collapse (whether that happens or not, the kind of technology that might be needed for such a doomsday scenario is being developed, and they're hoping to use it for the study of mental diseases and disorders in the not so distance future).
Oh, and before anybody else can get a chance: pr0n transcoding and transmission.
Processing our history, by looking at genomes, needs faster computers. It's taking months or years to process the DNA data that can be gathered pretty quickly.
Tons of scientific endeavors need to simulate their possibility before being worth trying. Likewise, industrial implementations need to do the same. CAD software, FI, keeps getting closer an closer to allowing the designers to see how it will act in reality, while whatever they are working on is still nothing more than a string of bytes in a computer.
Cheaper communication. 3rd-world countries are not only getting serviceable leftovers, but modern smartphones, tablets, netbooks, etc., that they can actually afford, and the internet is reaching them.
Then, there's ease of software development and maintenance. All this active web stuff is inefficient as all Hell, but it makes creating reliable software much easier, so more time can be spent on features, keeping up with regulations, or whatever else, rather then dicking about with segfaults, null pointers, and/or the prevention of them, in fairly low level language. But, that old P4 or A64 just can't handle it. Where I work falls into this case--so much of the important software is hosted services, relying on modern web browsers, and needs at least Core 2 level CPU performance to not impede the user.
Finally, entertainment, in the case of video games.
That's off the top of my head, anyway. For the most part, it's that there's consistently somebody that either can't do something, or can't do it well enough, without either more performance, or greater efficiency (in some cases, both, when either cooling or battery tech are the limitations). It's when the new tech comes out that the rest of us can find out about what they were waiting for.