Should I list competing offers on my LinkedIn profile?

First, congrats! This is huge, especially in this market.

Might I ask : Are you not planning on accepting any of those offers? Are you still waiting for an offer from a specific company? Also, would you mind sharing your interview timelines from (cold) application to offer?

Going back to your question, my personal opinion : I am not a fan of listing your offers as experience on Linked In, it does indeed come off as "braggy". But writing a couple of articles on Linked In would yield the same effect you want without the brag part, especially if one of them gets some attention. You could share about your experience, interview prep, or even write about some technical concepts you learnt in system design prep that you didn't know before (casually mention at the beginning and/or the end of your post that you learnt this as you were preparing for interviews, and that you landed several offers from Big Tech companies, basically find a way to include this info). Also, you could @ mention companies if you write posts on articles from their engineering blogs, this would increase your visibility.

Another thing you could do is reach out to recruiters/companies you're still in the process for and kindly ask to speed up the process because you got offers from X and Y company (mention the most prestigious ones). If you haven't had a phone screen yet, try to send it to recruiters on Linked In from the target company. Something like "Hey recruiter, I'm reaching out because I applied for a SWE role at your company. I recently secured offers from X and Y company for a SWE role (include level if relevant), but I am really excited about the opportunity to work at Company Z (+ Add a sentence for a reason why you genuinely want to work there) Is there any chance you could expedite the process or forward my resume to the hiring manager? I've also attached my cover letter/job id/portfolio for your reference. I appreciate your support. Thanks". Feel free to adapt it to your personality and style.

Hope this helps.

The only reason to put them all on is if they're all intern offers & you end up taking a leave of absence to do all 5 internships. Else, just put the one offer you end up taking.