Learn Portuguese (Portugal): Tenho de alcançar o auge, custe o que custar. - I have to reach the pinnacle, whatever it costs.

This is a reduplicative expression used to say "no matter what happens" or "at all costs". It is formed by taking a verb in the present subjunctive (custe), adding a relative word like o que, and then repeating the verb in the future subjunctive (custar). You can do this with many verbs, like diga o que disser (whatever they say) or faça o que fizer (whatever I/you do).

Is the second verb in custe o que custar just the infinitive?

It looks like the infinitive, but it is actually the future subjunctive! For regular verbs like custar, the future subjunctive is identical to the infinitive form. You can see this clearly when irregular verbs are used in the exact same structure, like seja o que for (where for is the future subjunctive of ser, not the infinitive ser) or venha quem vier.

Does o auge refer to a physical peak, like a mountain?

No, o auge is almost always used for a figurative or abstract peak, such as the pinnacle of a career, the climax of an event, or the height of an empire. If you are talking about the literal physical peak of a mountain, you would use o pico or o cume.

I sometimes mix up ter de and ter que. Is tenho de alcançar the only correct option here?