tonnenweise
tons of
Colloquial and exaggerated for a very large, countless amount of something.
Für die Party haben wir tonnenweise Chips und Getränke gekauft.
For the party, we bought tons of chips and drinks.

by the ton / tons of
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tons of
Colloquial and exaggerated for a very large, countless amount of something.
Für die Party haben wir tonnenweise Chips und Getränke gekauft.
For the party, we bought tons of chips and drinks.

The word consists of 'Tonne' (ton) and the ending '-weise'. This ending forms adverbs with the meaning 'in the manner of ...', so here literally 'in the manner of tons'.
Nach dem Sturm lagen ästeweise Blätter auf der Straße.
After the storm, leaves were lying on the street in piles.
'Tonnenweise' (by the ton/tons of) is an adverb and is never declined. Its form always stays the same, no matter which noun it refers to.
Im Herbst gibt es hier tonnenweise Äpfel zu ernten.
In autumn, there are tons of apples to harvest here.
In the meaning 'sehr viel' (a lot), 'tonnenweise' (tons of) is colloquial and a strong exaggeration. It is used in informal conversation to emphasize a huge amount.
Ich habe noch tonnenweise Arbeit zu erledigen, bevor ich in den Urlaub fahren kann.
I still have tons of work to do before I can go on vacation.
The literal meaning is used in technical or industrial contexts (e.g., with steel, coal, grain). The figurative, colloquial meaning is found in everyday language for almost anything (e.g., homework, money, problems).
Das Schiff transportiert tonnenweise Eisenerz.
The ship transports iron ore by the ton.
massive / in large quantities
'massenhaft' and 'tonnenweise' can both express a very large amount. 'massenhaft' is more neutral and fits things, people, or events. 'tonnenweise' is often colloquial and exaggerated, especially when you want to say that there is an extremely large amount of something. But it can also be meant literally when an amount is really measured in tons. Learners may confuse the words because both mean 'a lot'. As a practical rule: 'massenhaft' is more general, while 'tonnenweise' sounds stronger and often more casual, or refers to weight.

Für den Winter hat der Betrieb tonnenweise Streusalz gelagert.
The company stored road salt by the ton for the winter.
Vor dem Konzert standen massenhaft Menschen vor der Halle.
Before the concert, huge numbers of people were standing in front of the hall.