Being an indispensable employee is more important than ever because the economy has forced employers to reevaluate their overhead costs. You were probably hired because you were the best person for the job, but now your boss might be asking herself how essential your position is to the company. The goal is for your employer to believe they can't function without you, increasing your job security and (down the road) giving you leverage when there's opportunity for internal promotion or at bonus time.
A big part of the path to indispensability is becoming acutely aware of the people around you. If you know that your supervisor despises a certain recurring task and you're capable of taking it on, why not offer to incorporate it in your workload? Unless you have absolutely no time in your day to fulfill another small task, offering a helping hand can be very meaningful down the road.
This isn't about showing off your skills or proving how well you perform under pressure. It's about getting your supervisor to think you're a saint. Think it's going to have the opposite effect and your boss won't want to promote you? Think again. The task you've taken on will just become part of the job description for the next person hired to fill your shoes.