Genres of non-free software
Over my years of using free software (in the GNU sense) on Microsoft Windows, GNU/Linux, and now Android, I've found a few classes of software for which I don't see open source or free software taking over any time soon.
Games are one of them. A video game is made of both the "engine", which is a computer program, and "assets", an umbrella term for meshes, textures, maps, audio, and similar material other than code. The authors of high-quality assets still haven't adopted free culture motives to the same extent as programmers. Even if the engine itself is free, such as Id Tech 3 or Torque 3D, developers of the assets that sit on top of the engine still need to eat. What is an emulator without ROMs, or a Doom player without WADs, or ScummVM without the original data files developed by LucasArts, a Walt Disney company? And though selling support works for some kinds of business software, games that aren't massively multiplayer tend to need far less support from the publisher. Free software is best at making libraries and other software that programmers themselves use (jcnnghm) and whose scope a programmer can easily define; original games don't have that kind of clear scope.
Perhaps in a free software world, "there's no business case for creating videogames" (turbidostato, tehcyder, anonymous), and a developer should "avoid fields which prove to be unprofitable. You don't have to make video games, do you?" (alexo) Not everyone agrees though; some have suggested alternate business models, such as advergames or a guarantee of early access to new episodes (wrook).
For a similar reason, client software used to watch streaming videos published by the mainstream entertainment industry, such as Netflix software, will remain proprietary. The music and film industry associations of various countries (MAFIA for short) don't want people teeing the video into a local copy free of digital restrictions management, which would break the rental-like business model that Netflix and other video-on-demand providers use. So they impose robustness rules on cable TV operators and Internet TV operators such as Netflix, which rule out the use of free software in the video playback chain.
Tax preparation will also probably remain proprietary because "ABSOLUTELY NO WARRANTY", as several free software licenses put it, just doesn't cut it. The big U.S. tax preparation software companies (Intuit and H&R Block) stake their corporate reputations on the accuracy and timeliness of their translations into machine-readable form of the unending changes to the tax codes in dozens of jurisdictions. The comments to a Slashdot discussion about tax software might raise interesting points.
Lack of feature parity
And even where free applications are available, they often lack feature parity with the standard proprietary applications in a particular industry. Lumpy on Slashdot has found several applications that businesses can't do without as of 2013.
The fabled "year of the Linux desktop" is the year when these get ported.