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All answers having "Fat" as clue:

3 Character Answers:

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LIP



4 Character Answers:

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FLAB

GRAS

LARD

OILY

OLEO

SLIM

SUET



5 Character Answers:

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LARDY

LIPID

OBESE

PORKY

PUDGY

PURSY

STOUT

SUETY

TRANS

TUTOR



6 Character Answers:

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BUTTER

CHUNKY

FLESHY

GREASE

GROSSE

ORTHIN

TALLOW



7 Character Answers:

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ADIPOSE

OLEOOIL



8 Character Answers:

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ANDSASSY

SCHMALTZ



9 Character Answers:

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CORPULENT