In general, the least effective way to show initiative is to ask your boss for more work. The best way to show initiative is to find something that needs to be done and do it.
Everyone wants to be viewed as capable and trustworthy. More than anything, a supervisor wants to know that if she asks you to do something, you'll do it. She's looking for the assurance that she needn't follow up that simply putting you in charge is the same as having it done.
Here are some additional tips to help you demonstrate initiative at work:
Show up on time
It is hard to be credible in your claim that you are a self-starter if you can't show up to work on time. The earlier you show up, the more initiative you show. The boss knows what time the employees arrive, even when they arrive before she does. Bosses talk about this stuff.
Don't get caught slacking
Don't ever get caught playing solitaire at your desk or chatting at the water cooler when you should be working. If you want your boss to see you as someone who gets things done, you don't want to be seen doing nothing.
Report on your completed projects
Be sure to report on your completed projects to your boss. If you have her complete confidence, she may start to take their completion for granted, but until then, be sure to report when you finish your assignments. Once she's convinced that you'll always get things done, feel free to give her occasional reports, especially when unusual work or innovation was required.
Help a colleague
Everyone gets bogged down, at times - even you. When a colleague gets behind, offer to help. In some circumstances, you can jump in and help out. Your boss will likely be aware of the gesture. You may be glad you did when the shoe is on the other foot, and you need help with a project, whether it is because of your workload or a gap in your skillset. Build relationships around the office by helping whenever possible.
Accept new assignments
It may be the lowest form of showing initiative, but when your boss asks you to tackle a new project or responsibility, accept it. Then, execute on the new responsibility - along with your old ones - flawlessly. Don't negotiate for an increase in pay or relief from other responsibilities every time the boss needs something done.
Make a suggestion
Another way to demonstrate initiative is to identify a way the group's work product could be improved and offer to lead the effort to implement the new process. If the new process would fall entirely within the scope of your job description and authority - just do it. Don't ask for permission to do your job better. Only suggest the new process if the boss's approval or cooperation is required; then be prepared to execute.
You can become the person your boss counts on more than anyone else. As you know, the one boss counts on the most, often becomes the boss.