Since I was 12 years old I struggled with the choice between rechargeables and normal battery. I think we've progressed to the point where most discrete rechargeable batteries are not worth it anymore. I'll start with some stats and then conclude with some anecdotes. Typical use cases for me are flashlights, wireless mouse and keyboard, bike lights, alarms.
Most devices (phones, cameras, tablets) that are most suitable for a rechargeable battery comes with a good one, others are so efficient that it takes a long time to run them down (bike light, mouse, keyboard). A wireless mouse made in 2005 often need recharging every couple of weeks (remember the time when they come with a charger), nowadays, these things run for 6 months to a year without a recharge using the good/original batteries. Even my bike light, which I use every other day for 10min, took a year to finally run out of battery. On the other hand, things you use all the time that draws a lot of current like phones, cameras, computers, tablets, even flashlights often come with its own battery.
The first thing is self-discharge without any use. Here are some self-discharge rates for various batteries:
Li-ion: 8%/month at 21 °C
Basically, these suckers will now run out just by self-discharging long before when a normal battery died in the typical use cases these days.
http://batteryuniversity.com/learn/article/whats_the_best_battery provides a more detailed table and more properties.
Second, they have smaller capacities to begin with, often by a factor of 2-3.
1) rechargeable leakage is a problem that I'm sure many has experienced over the years.
2) Sometimes devices get lost, so your investment is not worth it even if it would have in an ideal world. After my bikelight ran out of battery (after a year of regular use), I replaced them with some rechargeables. In 2 weeks, they are already barely lighting up, and after another 2 weeks, the light got stolen.
3) Environment: one of the better rechargeable, NiCd is also the most environmentally harmful. It is already partially banned in the EU.