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Some context:

I've worked at my current company for about 2 years now as a software developer. It is a medium to big company with a solid grasp on the market. Also, it is a really cool workplace and is pretty pleasant in general, except the pay. My initial salary was about 30% lower than the average where I live. Recently, I started looking for other job opportunities and got a offer to join a startup. The salary there is right around the average, so 30% more than my current salary.

With this offer, I approached my manager and told him that I was considering leaving, and that the money was the decisive, and only, reason. He quickly asked for a number and said he would get it done by the end of the week, which he actually did. He said that I was a very valuable member to the team, and that they could not afford to lose me for something so small as a raise. During the weekend though, I received another offer from a different company, and the salary is around 30% more than I get even with the raise. The company is regarded as the best tech company to work for in Brasil, and I'm considering them as an option.

Will I burn a bridge by leaving right after asking for a substantial raise so I would not leave?

Simply state,

"Boss, you wouldn't believe it. I'm making 6. You kindly offered me 7. Out of the blue on the weekend XYZ offered me 8! No kidding - here, you can see the offer. I want you to understand I'm NOT trying to play you for more money. I am definitely taking the "8" offer. Thanks for the great years! Have another Caipirinha man!"

End of story.

This is totally and completely uninteresting in the world of software. No issues.

As you say you don't want to "be rude" so just make it totally clear that you are NOT looking for yet another offer.

Your old boss will appreciate you being decisive, truthful, frank and crisp.

Done!

Communicate.

Talk to your manager and explain again that you have had an even better offer. Show him proof if needed (and willing of course). He may say he'll match it, you may be able to negotiate higher e.g. 10% but you still have to make the choice, Is that 10% more important or is joining

The company is regarded as the best tech company to work on Brasil

You need to decide whether you enjoy your current job/role enough to stay and potentially negotiate an even higher salary. Or get this massive opportunity on your resume. With your statement above I assume this company has good reputation and is highly commendable as a company.

To me, your current company will not likely increase your salary above the offer. Maybe match it, but not top it. Then you also have the reputation of this company behind you. To me it's an obvious choice on what I would be choosing.

You won't be burning any bridges at all, your manager will understand why you have made the choice. If they can't beat the offer they have to let you go. Your manager clearly thinks highly of you so you will still have that strong relationship between yourselves if ever required for whatever reason.

You might burn a bridge in this case.

You gave your manager a specific number so that he could retain you ( you mentioned to him that money is the only reason for you considering leaving ). He then promised to give you exactly what you asked for and delivered on his promise ( who knows how many hoops he had to go through to fulfill your request ). And now you will go to him and tell him that money actually wasn't the only thing that mattered.

I can see some managers being upset at losing an employee in this manner.

Regardless, I don't think that burning bridges matters much in this case. A substantially better opportunity has come along ( best tech company and much better pay ) and those opportunities don't come around frequently. I would take that offer even if it means burning a bridge with your current manager. He may be upset, but at the end of the day he will understand that you made the right decision.

Your company was significantly underpaying you, for years, and they knew it. When you pointed it out, they offered you a raise, which is (kind of) nice of them (if you overlook all you lost before). But now you have an opportunity to be decently paid from the start. Take it!

In any case, staying after setting up a bidding war over salary rarely comes to a good end, you're now marked as a flight risk and your career in this place is inherently limited. If you don't go now, chances are they'll push you out in fairly short order. And if you still love them after you work elsewhere, you can always come back, for even more money, later, that's a different setup. But once you show a higher offer to your employer you pretty much have to leave.