I have a storage shed that has a twist timer so that the lights turn off every hour. Other than replacing or removing the timer (I don't have authorization to do that nor could I turn off the electricity to get it done), how can I get the timer to stop, so that the light stays on until I turn it off?
The timer was put there by the storage owner so that their electric bill isn't subject to abuse by people renting the units. It also prevents (for instance) people using the units as marijuana growing facilities, or other kinds of production (legal or otherwise) and getting the storage service to pay the electric bill. If there isn't an unswitched outlet in the unit, there isn't a "hack" to bypass the timer, and as you note, altering the wiring would surely violate your rental agreement.
If your need for long term power is low-draw, you may be able to rig something using batteries and an inverter, but you'd still have to supply the power to charge the batteries, or put a screw-in outlet in the light socket to providing charge on the timed power (and be limited to the amount of charge you can put into the battery in an hour at a time). As Robert Cartaino noted in comments, if light is all you need, a suitable battery lantern is probably a better solution than trying to find a way to defeat the light switch timer.
It is somewhat jarring to have the lights suddenly go off when you are in the middle of something, but the timer is necessary to ensure the lights aren't left on long after you are gone.
If you want to ensure the lights stay on without disruption, set a recurring reminder for 55 minutes on your smart phone (or kitchen timer) so you can reset the light timer if a longer duration is needed. That will give you a bit of time to turn that twist-timer back when it is convenient… rather than getting caught stumbling around in the dark when the lights go off unexpectedly.