Which country? Common practice can vary WILDLY across the globe. I've stopped even trying to answer questions about hiring and salary and company culture in India, for example; my US-based answers would be laughably wrong.

If you get an offer from another company that's better than what you have, why wouldn't you just take it?

Leave and come back. You won't be the first.

Interview, get offered a 25% increase, get current employer to match: total gain 25%. Interview, get offered a 25% increase, accept offer and leave, get (former) employer to offer you another 25% increase to come back: total gain ~56%.

Have they given you any reason whatsoever for not giving you a raise to get your salary in line with your experience and market value, or are they just flat refusing to do so? Especially since they acknowledge that you're significantly underpaid?

Leave, they clearly don't respect you enough to pay you what they know you're worth. It's as simple as that

Did they adjust your job description along with the increased salary? In some companies, there is a maximum salary for each position...

Expect less than inflation rate? That's kinda crazy.

Does this answer your question? How should I properly approach my boss if I'm feeling underpaid?

Is it in any way practical to apply to another role at your very company through the process normal for new hires?

@SethR: Maybe he likes his current job?

You should definitely get an offer letter from another company, if that's what your employer wants. Then accept the offer, and once you have a start date for the new role - quit your current one!

"I know, I don't really want to change roles/company and start over whatsoever just to have a nice monthly pay but quite possibly a worse role." — that's why they don't need to increase your pay.

That other place you got an offer letter from just to be able to show it to your boss, with no intention of accepting it, may (may!) be unhappy at being "used". Attending interviews "to stay sharp", if you reject an offer, and then apply again later - wouldn't this harm your chances of being hired? However, as I wrote there, you might just find out that the other company actually is a better fit, not only financially. You will never find out unless you go and interview.

Find a new job with better pay and leave as soon as you can and don't look back. This company (and particularly the branch manager) has no respect for you and are just exploiting you. They admit they pay significantly better to juniors yet they want to make you jump through hoops to "prove it". Also, never accept a counter offer.

The question is if you are willing to stay if you don't get a raise. Next question is how much you are willing to do on your own, to get the raise. Think it carefully through because those answers will most likely then show you what is important to you.

After update: Wait, what? They're increasing your responsibilities and still paying you crap. You are being used - but I'm starting to think you like it.

IANAL but this is silly... You just need Offer letter - I can create dummy ltd, write you offer for bambilion $ and dissolve the dummy ltd. You have your letter (which is just piece of paper bcs company no longer exists) and I will never pay you a single penny - you can sue but good luck, this ltd (which was legally responsible, not me.) no longer exists...

What would their reaction be if you just refused to negotiate that way?

I'm wondering if the company policy limits the amount of raise they can give yearly, but you did not apply for each year, so you've lost 6 years of raises ? In all cases, even if you like the job, you're not well anymore with this company, you should move.