Fòram na Gàidhlig

Bha mi leughadh.
I was reading.

nighean mic nighinn bràthar m' athar = my paternal uncle's daughter's daughter?

Almost - daughter's son's daughter.

These long lists of words in the genitive case only occur with family members.

Otherwise you only put the last word in the list or chain into the genitive, and also only the last word can have an article.

iuchair doras taigh ministear eaglais a' bhaile.

Work that one out!

It might help to think of a remarkably similar - but toally inverted - construction in English, e.g. "The town's church's minister's house's key". This is a mirror image of the Gaelic, in that (Gaelic) only last noun genitive <=> (English) all but last noun genitive; (Gaelic) only last noun has article <=> (English) only first noun has article - and, of course, the actual order of the nouns is exactly reversed.

So, just as you wouldn't say (at least, not in any dialect of English I'm familiar with) "The town's the church", so you wouldn't say in usual Gaelic "An eaglais a' bhaile". In both English and Gaelic, the "church"/"eaglais" bit has no article. So you can think of a general rule: "If I wouldn't put an article in in English, using the "'s" genitive construction, then there's no article in Gaelic".

Ceartachadh beag: Bha mi a' leughadh.