This is a very interesting phenomenon. It was certainly true from around 1968 to 2005, that custom chips could not survive since the next Intel chip made them irrelevant. But I think that, not only is this going to be changing soon, in fact it has already started to change.
There are credible claims of BLAST (and other life science codes) being ~10x faster on GPUs than CPUs. [1] GPUs are specialized processors (with associated non-standard programming models and compilers) that have overtaken general purpose CPUs in many areas. This has been possible because, for the past 10 years, the "next Intel chip" has NOT been faster (single core clock speed) or exponentially cheaper (cost per core). Intel has been unable to make advances on these fronts due to fundamental engineering factors, such as the breakdown of Dennard scaling around 2005. [2]
GPUs (and FPGAs, and to some extent even conventional CPUs) have been able to continue advancing since 2005 because Moore's Law has been holding (even though the related Dennard scaling law has broken down). However, now we are seeing that even Moore's law is starting to break down. This is evidenced by the increasing delays in the Intel roadmap, with the most recent delay being 10nm process being pushed out to 2017. [3] It is now in question whether silicon process will ever even reach 7nm, and I don't think anyone is willing to bet that Moore's Law can continue on silicon beyond 7nm.
It is at the end of Moore's Law where custom chips become very interesting. This has been covered in 2013, in a presentation by Robert Colwell, who is the Director of DARPA's Microsystems Technology Office. [4] Colwell's thesis is that the End of Moore's law will revive specialized chip design. I find that prospect to be very exciting, not only for chip designers but of course also for software developers (especially compiler and programming language designers, as specialized chips are going to require specialized compilers, programming models and languages).
Of course there is also the other possible future path, within the 10nm to 7nm time frame (i.e. within the next 5 years) that Intel and others will find a way to extend Moore's Law, possibly by finding a viable alternative to silicon substrate. That would also be extremely exciting but somehow I don't think the future will be that simple (DARPA seems to take the view that this simple isn't going to happen, i.e. the Moore's Law exponential must logically end).
[1] https://www.nvidia.com/object/bio_info_life_sciences.html
[2] https://en.wikipedia.org/wiki/Dennard_scaling#Recent_breakdo...
[3] http://www.anandtech.com/show/9447/intel-10nm-and-kaby-lake
[4] http://www.hotchips.org/wp-content/uploads/hc_archives/hc25/...