I am currently using Ubuntu and I appreciate the big community, the Software Center and the fact that it's very easy to use. But additionally I'd like to have the latest updates, like e.g. LibreOffice 3.4. Which distribution should I try?
Here are some of the major distributions ordered from "newest" on:
Arch Linux and Gentoo are the two most popular rolling release distributions. This means they will always be very new, even "bleeding edge", and this includes core components such as X and the kernel. Both will require fiddling with your system and it's recommended to be familiar with Linux internals (Arch might be slightly easier to get into, though).
openSUSE has Tumbleweed, which is an attempt to create a rolling release distribution from openSUSE. It aims to support only stable releases and hence should be more stable than Arch and less "progressive" than Fedora (that said, I haven't tried it; I use vanilla openSUSE). Notably, they don't support proprietary graphics drivers (due to frequent kernel updates) so don't even think about it if you use t.
Fedora is often mentioned as a "bleeding edge" distribution. It's not as bleeding edge as Arch/Gentoo, of course, but it's ahead of the other major players. With Fedora this also often means architecture updates, which means that updating to the next release can be painful (eg. GNOME 3, systemd in this cycle; btrfs as the default filesystem next cycle). Still, you will have newer applications in general.
Ubuntu. With a six-month release schedule, Ubuntu actually updates more often than many distributions. Still, sometimes it will not have the latest software (as for LibreOffice for you). However, due to Ubuntu's popularity it should be easy to find a repository with most major applications, while you can keep the rest of the system as is.
Everything else... :) (I stopped at Ubuntu as that is your point of reference)
It now boils down to what exactly you want. If you wish the newest versions of specific applications (eg. Firefox, LibreOffice, Chromium), I'd advise you to stick to Ubuntu. This is the safest option. If you want to have all new applications, Fedora or even openSUSE:Tumbleweed might be a better idea. You will have problems, but it will generally work. Finally, you can go with Arch. It'll require setting up, and then some effort every now and then to keep it all together, but you will be very bleeding edge. Arch is also good if you want new versions of some more obscure applications (eg. GRAMPS), which might be lower priority in other distributions.
All of these have a big, vibrant community around them so you should be able to get help for any issues you face. Still, the "ease-of-use" scale is approximately the inverse of the above list, so you should weight the options against that.
If you want current packages, then it is empirically proven that Arch Linux is the least obsolete distro with an average lag of 8 weeks between upstream and standard repos.
The next most current distribution is Fedora, with an average lag of twice that, 16 weeks.
The closer you get to the most recently released software versions the more problems you will run into. That being said most of the major distributions play leap-frog as they move forward, each new version passing up the competing distributions. The answer to this question will change depending on which week you ask. Ubuntu tends to be pretty cutting edge. I'm sure you can get LibreOffice 3.4 on it from some new repo.
Some distributions allow you to do more of a rolling upgrade where major new versions of softwre are constantly put into the repositories. You might find something like Gentoo useful in this department. My favorite, PLD-Linux, has a continuous release tree. I think Arch Linux also works on a rolling release model.
Fedora is pretty aggressive. Their release often has newer versions of many non-core components than Ubuntu, particularly of things outside the Ubuntu main repository. Rawhide is the development version, comparable to Debian Sid (Unstable) and tracks things pretty quickly.
Debian Sid is fast-moving in some respects, but it can take a long time for major components such as a new Gnome release to make it in (or at least to filter down to Testing).