I lived in Japan in the 90s and remember the (big) fish section in the supermarket. A label on one said, if I recall correctly, "abura no tappiri" which I believe means 'filled with fat.' Is that correct? And is that considered a bragging point for fish? (note: I never saw that on pork or beef)
脂【あぶら】たっぷり (I believe tappiri is a typo for tappuri) is a phrase meaning "fatty", "has put on a lot of fat" (but "filled with fat" is probably an overstatement). It's regarded as a good property of tasty fish. Fat-rich parts of tuna even have a distinct name (トロ). For beef, 霜【しも】降【ふ】り ("marbled", "fat-laced") is more common as a marketing phrase.
Note that 脂【あぶら】のたっぷり makes no sense in isolation. Perhaps you remembered it incorrectly or you remembered only a part of a longer phrase.