A level 20 barbarian gets unlimited rages. Can a Berserker barbarian extend his rage by raging again endlessly so that it never ends - which would lead to him never gaining exhaustion and never losing Frenzy?
This is likely not possible, some interpretation is required
The Berserker Barbarian's Frenzy feature states:
Starting when you choose this path at 3rd level, you can go into a frenzy when you rage. If you do so, for the duration of your rage you can make a single melee weapon attack as a bonus action on each of your turns after this one. When your rage ends, you suffer one level of exhaustion.
The Barbarian's Rage feature states:
On your turn, you can enter a rage as a bonus action...
Your rage lasts for 1 minute...
I would argue that you cannot actually enter a rage while you are already raging and thus you could not activate a rage during another rage. This is also supported in the Rules As Written interpretation of the most upvoted answer to the question "Can I start a new rage before the previous one ends?"
Regardless, even if you could enter another rage while currently raging this would be a new rage. The rages wouldn't stack because they are game features of the same name but their durations would still count down. Therefore your first rage would still end after its minute had gone by and thus you would gain a level of exhaustion from Frenzy.
I believe you can't "extend" your rage at all, but even if you could enter a rage while currently raging, you wouldn't actually be extending it --- you would be starting a new one and the old one would end at its usual time.
One small interpretation requirement:
The Frenzy feature says "when your rage ends" but if we can have multiple rages at once we would have to interpret this as "when one of your rages ends" for the above analysis to work. If it instead meant "when you stop gaining the benefits of raging" then your strategy of never stopping would work. But presumably the exhaustion level from Frenzy will apply even if there are two simultaneous rages and one of them ends.