Seriously, I can't think of an example and I've used hundreds of audio-playing apps.
OSS apps? Use osspd.
ALSA apps? Use the pulse plugin for libasound2. Installed by default and configured on all modern distros. Bugfixes galore have hit it in recent months/years, so if you aren't using the latest stable distro (and NOT Debian Stable, that doesn't count), either upgrade or stop whining.
OpenAL-soft apps? Solved problem; works well with native pulseaudio API as backend.
SDL apps? Solved problem; works well with native pulseaudio API as backend.
Proprietary apps? Flash, Skype, Savage 2, Heroes of Newerth, HIB games, VMware Workstation, Second Life all work out of the box... do I need to continue?
JACK apps? Start jackd and run PulseAudio on top of it, problem solved. AFAIK Ubuntu and Fedora are working on a way to get JACK to automatically start-up and run PA on top of it whenever something needs to use JACK. It'd certainly be nice to have it automatically just work, but hey, if you're using ALSA or OSSv4, you're used to inscrutable configuration files already.
And let's not forget that almost any software you can download from a distribution today is working with PA out of the box. People test this and file bugs when things aren't working. So the primary cases where PA doesn't work out of the box are going to be third party apps, and let's face it, third-party developers can do whatever the fuck they want, even if they explicitly desire to make their app as incompatible as possible with PulseAudio by statically linking to a compiled-in libasound2, initializing ALSA with snd_pcm_open("hw:0") and encrypt the string in the binary and use an executable packer so you can't easily binary hack it to open the default pulse device either.
This isn't a new problem. ANY platform that allows arbitrary third-party machine code to execute is going to have incompatibility problems. I can't tell you how many times I've run a Windows program and watch it BSOD my machine or wreak similar havoc. The hope is that most software that people will ever want can be provided by the distro, so that the benefit can be enjoyed, of making sure the distro integrates everything and ensures that it all works together. We're not 100% there yet, but PulseAudio is the last best hope for Linux audio. If you don't want Linux audio to graduate out of the 90s, you're welcome to run Debian Stable or RHEL5.