Learn Hardcore German: Ich verbrauche höchstens drei Liter Wasser beim Kochen. - I consume at most three liters of water when cooking.

höchstens = “at most,” “no more than,” setting a hard upper limit: you won’t exceed that amount.
Example: Ich verbrauche höchstens drei Liter Wasser.
maximal is a near-synonym, a bit more technical/formal: Ich verbrauche maximal drei Liter Wasser.
bis zu = “up to,” often used for ranges/advertising; it suggests a ceiling but less of a firm personal limit: Ich verbrauche bis zu drei Liter Wasser.

Where can I put höchstens in the sentence?

It’s a flexible adverb. Natural options:

Ich verbrauche beim Kochen höchstens drei Liter Wasser.
Beim Kochen verbrauche ich höchstens drei Liter Wasser.
Höchstens drei Liter Wasser verbrauche ich beim Kochen. (fronted for emphasis)

Putting it right before the number is most common. Avoid splitting it oddly from what it limits.

Why verbrauchen and not brauchen, benutzen, verwenden, or aufbrauchen?

verbrauchen = to consume/use up a resource (water, electricity, fuel). Best fit here.
brauchen = to need. Ich brauche drei Liter Wasser means “I need three liters,” not “I consume.”
benutzen/verwenden = to use (utilize). You can say Ich verwende/benutze beim Kochen Wasser, but it doesn’t stress consumption the way verbrauchen does.
aufbrauchen = to use up completely (finish what you have): Ich habe die drei Liter aufgebraucht.

What exactly is beim Kochen, grammatically and meaning-wise?

beim = contraction of bei dem and takes the dative.
Kochen is a nominalized infinitive (a noun), hence capitalized.
Meaning: “while cooking/during cooking,” i.e., during the activity.

Could I say während des Kochens or während ich koche instead of beim Kochen?

Yes:

während des Kochens is a bit more formal.
während ich koche is a full clause (very clear and neutral). All three mean roughly the same; beim Kochen is the most colloquial/compact.

Why is there no article before Wasser in drei Liter Wasser?

Because it’s a mass noun in a measure phrase. German typically uses the bare noun after a measure:

drei Liter Wasser (natural, general) You’d only use a determiner for specificity:
drei Liter des Wassers (very formal/specific)
drei Liter von dem/vom Wasser (colloquial/partitive, from a specific source)

What case is drei Liter Wasser in, and which word shows it?

It’s the direct object in the accusative. The case shows on the measure noun:

Ich verbrauche einen Liter Wasser. (accusative: einen Liter) With higher numbers, the article disappears: zwei/drei Liter Wasser. The noun Wasser stays bare.

How does Liter behave in the plural and in different cases?

Singular: ein Liter
Accusative singular: einen Liter
Plural (counting): zwei/drei Liter (no -s)
Dative plural adds -n: mit drei Litern
Genitive singular (rare in speech): des Liters In your sentence it’s accusative plural: drei Liter.

Can I re-order the sentence elements?

Yes, keeping verb-second:

Beim Kochen verbrauche ich höchstens drei Liter Wasser.
Höchstens drei Liter Wasser verbrauche ich beim Kochen.
Ich verbrauche höchstens drei Liter Wasser beim Kochen. All are correct; choose based on what you want to emphasize.

Does kochen here mean “to cook” or “to boil”?

In beim Kochen, it means the general activity of cooking (preparing food). Kochen can also mean “to boil” in other contexts (e.g., Das Wasser kocht = “The water is boiling”), but not here.

What’s the opposite of höchstens?

mindestens = at least (numeric lower bound): Ich brauche mindestens drei Liter Wasser. Related but different: zumindest/wenigstens (“at least” in a concessive sense) don’t set a numeric minimum as directly as mindestens.

Any quick pronunciation tips for tricky parts like höchstens and verbrauche?

höchstens: stress the first syllable; ö is rounded (like French eu), and ch is the soft “ich” sound.
verbrauche: au like English “ow” in “now”; final -che has the same soft ch.