I'm not interested in software that's been patched over the years, but code that's been unchanged for years and is still in active use. Are you using any DOS programs, for example?
 And no one individual programmer should be solely responsible for life-or-death systems or space-bound systems.
I'd define "in use" to mean still in productive service to a group of people or institutions significant enough to represent a niche market.
For that, I'd probably look to things like avionics software in certain airplanes, navigation systems on old ships, firmware in medical devices and diagnostics, or perhaps to inventory management systems in old warehouses.
And who knows what firmware we loaded onto our Cold War-era ICBMs, for that matter? Or what dusty, backwards-compatible systems we would need to maintain control over them? Would anyone like to play a game? :)
Just so long as the old compiler worked well enough, they had no reason to upgrade. (I did uncover a bug in the compiler which I had to work around, but wasn't enough to warrant an upgrade.)
--edit-- I also play Commodore 64 games in emulators, and they date from the 80s, but I suspect this is not the sort of answer you're looking for.
I took a look into it and the oldest macro I could find that I use semi regularly is SYS1.MACLIB(DCBD). A utility for working with data control blocks which dates back to 1977, putting it at 36 years old.
There are definitely older programs on the system, but I'm not sure how often they're used.
I don't know how old the application is, but DOS/VS is from 1972, and depreciated since 1980 by DOS/VSE.
It must have been about '08 when I did some work porting parts of Tivoli's event monitoring stuff to z/Linux.
So indeed, the oldest software will probably be running inside a VM.
It covers PDP's and some home systems (eg Apple IIe) too.
I have a zip of the download, and just keep installing it and moving forward. I never quite made the jump to Winamp 3 and once I stopped moving forward every update to Winamp deterred me some more.
Over the years I have tried everything, but I like WinAmp 2.
WinAmp 2 just works great. It's no nonsense and I love the quick find (CTRL+J) on the library... search file name, path, and metadata. Which is how I need search to work, based on my file conventions as well as the metadata.
Where I work we have hundreds of engineers programming in Teradyne's LASAR software, rehosting old programs onto new digital test platforms. LASAR was developed in the mid 80s and we have it running on VAX and UNIX servers. Since electronics no longer use combinational logic in discrete chips, no newer software is available.
The boring majority of software in the lab is just the latest iteration of a 'Microsoft-whatever' suite stuffed into a standard enterprise Dell box.
It's the exceptions that are interesting.
For example, a few weeks ago the lab's last PC that ran Windows 95 died. That really sucked for a few people because a piece of proprietary software that ran an important microscope camera only ran on Windows 95. (Only ran on Win95 if you didn't want to pay big time for an upgrade to a modern OS.)
Intriguingly, a PI I am working with is planning on using his old Silicon Graphics machine very soon. He told me there are still certain features lacking in modern software that can be found on software written for some older SG machines.
Finally, we are in the process of restoring an old machine that is frequently used for pulling apart very thin pieces of glass for electrophysiology experiments. Granted, all the software inside is embedded. But the software and the hardware still exceed 13 years of age by a wide margin.
Edit: Changed "in-virtuo" to "within virtual machines" for clarity.
The programs convert the ebcdic files on tape to ascii and then build a cheesy search index for other DOS based programs. It takes 2 different virtual machines, two different dos releases and 4 different programs to get it done.
The whole process is an exercise in Seuss-ian ridiculousness.
I still tease him about it.
List of missing things: EVERYTHING. Primary/Foreign key relationships are buggy, therefore, not used. Views are somewhat available. Joins are buggy. Have to build all indices by hand choosing the relational columns and data structure/allocation. Cannot subtract sets without huge where clauses with very odd syntax. Rely mostly on WHERE EXISTS/NOT EXISTS or WHERE IN. Can only perform single column queries in a where clause, no WHERE (a, b) IN. Its free and it works. IT is a cost center. No new databases anytime in the near future.
Although now that I look at it, they say my version of php will be phased out August 1'st :( And they are raising the price on August 1'st ...
If you click something that goes to page-not-found, make sure the link has 'orig' in front of it, because in some places I went directly to www
Aloha Restaurant POS
Watch this youtube video, then watch what servers use at your favorite restaurants.
I used it back in 1998 at a restaurant and it felt old then. Still used at almost every bar, restaurant I go to.
It was patched for Vista, but doesn't run fully under any x64 version owing to it's 16 bit code.
There is a new version in the pipeline at the National Council of Architectural Registration Boards, but the old validated test engine is still in everyday use.
The hangup is the Media Viewer 1.3 technology. I believe based on comments I previously  made elsewhere that this was Windows 3.0 technology - it's pretty much impossible these days to do a native search for anything which can be interpreted as related to a current commercial technology.
The practice programs for the Architect Registration Exam still use it and can be downloaded from NCARB:
Some of the "program" in AO-7 is discrete hardwired. Depends if you demand its reprogramable. In which case mask programmed rom devices are unintentionally removed. Or keyboards which are debounced in hardware instead of software are unintentionally removed because the debouncing is done in hardware logic not software. AO-7 is probably "close enough to count" although I can sympathize with an argument against it.
"The spacecraft included two command decoders and a command distribution unit, a very limited form of processor, to direct operations on the spacecraft"
If we call what it has a computer, it was reprogrammed remotely:
"Much of the computation for the mission was performed on Earth and transmitted to the probe, where it was able to retain in memory up to five commands of the 222 possible entries by ground controllers"
That happened as late as March 2001 (and possibly even later, as communication was finally lost in January 2003): http://www.strobedata.com/home/pioneer10.html
Voyager probes are still up and running but the power system has only perhaps one decade of useful life remaining as it declines.
I know a lot of old pub quiz machines still running on Windows 2000.
However, the real question here is: Why are you using a piece of software that is 13 years old even though there are patches/updates for it?
The utilities (include the telecom companies too if you want) have a lot of assembly code written ages ago that they're afraid to touch. I suspect that's worth a good first look. I think 50 years is a reasonable guess.
How about our very own jgc running calculation jobs from Charles Babbage?
If that's too historical then the chances are good it is written in Fortran, and being used by a cosmologist on a telescope somewhere up high.
That is, of course, if we're putting aside some of the basic UNIX tools that probably haven't changed since the 1970s—which I think is fair.
(A better example would be the bsdgames collection. It's a lot older and has barely been touched over the years.)
I looked at some other man pages from ls, cat, tar and the like and the oldest that I found was rm, from January 28, 1999.
UPDATE! rmdir's man page is from May 31, 1993! Though I'm sure the actual code was updated since then.
The core program was unchanged, there were just some minor changes for this specific application (e.g. the path changed, and there was like one if block added for this purpose). So it was pretty much unchanged for over 15 years.
Let's put it this way: there's a market for teaching people how to program like it was 1975. Nobody around can do it anymore, and the code base is large enough that both the teaching and skillset are highly valued in some circles.
I used 90s versions of Winamp and Xnews until I just stopped caring up local MP3s and USENET.
bfm is 'bubblefishymon', a system monitor that packs a lot of info into a picture, being that of a fish tank. The level of the water represents RAM usage, the amount of bubbles indicates CPU usage, fish moving leftwards is network traffic in, fish moving rightwards is network traffic out, and there's even a floating rubber duck... which represents a duck!
I know many who use many years old DOS based Tally, accounting software.