Off the top of my head, some of the things that have broken at an LTS transition that I've been involved with are out-of-tree kernel module builds, C code using OpenSSL, Puppet config, Salt config, RPM specfiles, Python code, Perl code, Apache configs, shell scripts, Java code, bootloader configs, bootstrap scripts, and init scripts/configs (esp. sysvinit to systemd). Any one of these things is not a problem in isolation, the problem is due to having to fix all of them all at once. Too much complexity put into any one of them (often arising from external requirements or rushed implementations) also makes migrating harder. Waiting until the 11th hour on the EOL clock just adds to the stress of the process.

Many of my bad experiences were because of corporate policies and lack of proper prioritization at levels above system administration. However, the sysadmin does have some choice in the matter, especially when greenfielding. You can turn stability into a vice if you're not careful.