When starting out as a programmer in the big, bad world life can be pretty daunting as is landing that first job. Trying to prove that you are not just a baby that needs a sitter was probably the biggest obstacle for myself. This will of course depend on the workplace into which you are hired, however from my experience talking with the support network it is one of the most common starting issues. Interestingly, in my experiences I generally noticed that juniors will wait until they are invited to prove themselves on a big task or project instead of actively seeking the opportunity to do so.
I should preface this notion with a disclaimer: if you are in your first 6 months of your very first job - probably don't go looking to prove anything. Once you feel that you've learnt everything you can in your current role (at least 6 months, normally closer to a year) then it's time to start looking for more advanced projects. The project shouldn't be on things that are completely foreign, rather subjects that you have some level of knowledge or understanding of but haven't been able to attempt in a professional setting. Maybe that's a job of a colleague who is always complaining they need help or that they have too much to do, or perhaps a new technology in your current field has caught your attention and you'd like to try your hand at it. These are perfect examples of opportunities waiting to be taken.
But how to ask? Again some workplaces are going to be worse than others when it comes to asking for opportunities. If you have the option to just ask your next-in-charge for a change of pace or a challenge then definitely take that path. Simply asking outright can save a lot of time and clearly communicates your intentions. If that isn't a choice then there is one technique that has worked particularly well for me, it goes like this:
A manager, supervisor or even colleague comes up and says something along the lines of
"I need you to do [the thing you always do]."
To which you reply
"I can do anything."
Wear a slight smile and look them deadpan in the eye. If they reply with anything that questions the comment (such as "is that so?") then you have succeeded, continue with "I can do anything, just give me time". Depending on the culture you might have to expand that to "just give me the time to do it". Sometimes however they will not question it and simply state something like:
"Great! well, I'll need that by COB."
Once again you need to reaffirm that you are ready for something else, so retort with
"Is that all? I thought you were bringing me a challenge!"
Even if the work they have given you is too much or will be a challenge to complete by the deadline, still reply the same. If you are not learning anything new from what they are giving you then they haven't provided you with a challenge, they've just given you more grind.
The point of these responses is not to come off as an egotistical know-it-all or to ooze arrogance. This point is to advertise that you can tackle something bigger or different. You definitely will not know how to do every part of the assignment that you are asking for. As a junior developer I can assure you, you don't know squat - but that's completely beside the point. As far as I am concerned no one in IT knows how to fix everything, however anyone in the IT sector worth their pay knows how to learn to fix their problems. This is exactly what you are asking for: time to learn how to fix different problems within the company - the phrasing is just to get their attention. Like verbal clickbait.
Repeat the "I can do anything" process as much as needed. The longest I've ever had to hold out was 2 and a half weeks. You will be broadcasting quite confidently that you are ready for something else and if nothing changes within a month - start consciously looking for another job.
I can't stress this enough, if you are not learning in your job and you are looking to make a career in IT then you are actively hurting your professional development by staying where you are. In IT you must always be learning. Keep in mind there is a difference between there being nothing in the work pipeline that fits what you are looking for and the company just outright dismissing the request as "outside of your skillset". If they open the communication up around the topic by stating that there currently isn't anything that they can provide you with but when something does come up they'll send it your way - then that's also a win.
Just ensure to take the comment with a grain of salt and keep saying you can do anything. Some of my friends have experienced similar situations, except they never recieved the challenge they were promised or any mentorship and unfortunately ended up leaving their company for a better position elsewhere some months later. If you're in such a situation and the company is big enough, I reccommend asking for a mentor or to pair-program with a more senior programmer. Really try to drive home the notion that you are hungry for the knowledge.
Of course if you are really lucky and work for a place that intentionally challenges and upskills their staff then you might never have to ask for such an opportunity. If that's you, count your blessings -- you are one of the lucky ones that has not yet experienced the wonderful soul-sucking dredges of corporate life at some companies. The reality of the world is that not all places are equal and until you end up in an ideal workplace keep shouting that mantra: "I can do anything", just watch me.