Up to now we've mostly been discussing indefinite noun phrases that are made up of three nouns and where the second and third nouns form a semantic unit. But consider situations where it's the first and second nouns that form a semantic unit. Here there will always be a double genitive i.e. the second noun will always be declined in the genitive e.g. [teach pobail] baile.

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Two possible arguments for leniting "baile" in the case of [oifig phobail] baile:

1. If it was simply "a town office", you would say "oifig bhaile" since "oifig" is feminine. And with [oifig phobail] b{h}aile, "baile" still qualifies the feminine noun "oifig", Granted, it does so only indirectly or, more precisely, it qualifies the noun phrase "oifig phobail". Maybe it's a case that, while individual nouns have gender, it makes no sense to talk about the gender of a genitive phrase, such as "oifig phobail", and so "baile" would go unlenited. On the other hand, you would say "oifig bheag bhaile" (a little town office) and not "oifig bheag baile", even though the word "bhaile" does not come directly after "oifig".


2. The word "phobail" which immediately precedes "baile" is in the genitive singular form while ending in a slender consonant, and so one interpretation of GGBC section 4.14 might suggest that "baile" should be lenited.

Otoh, the following two points would tend to rule out this second argument:

⦁ Relevant part of GGBC 4.14: "Séimhítear tuiseal ginideach ainmfhocail éiginnte...atá faoi réir ag ainmfhocal eile...nuair a chaolaítear san uatha consan deiridh an chéad ainmfhocail trí infhilleadh." (The genitive case of an indefinite noun that is governed by another noun is lenited when the final consonant of the first [presumably the governing] noun is slenderised in the plural through inflection.) However, in the case of [oifig phobail] baile, baile is not governed by "phobail". Instead it's governed by "oifig", or more precisely by the noun phrase "oifig phobail".

⦁ In GGBC 4.14, the only examples given have the second and third nouns as a semantic unit, as opposed to the first and second: toirtín [aráin choirce]; sárú [gealltanais phósta]. (As an aside, these examples curiously use the double genitive.)

In conclusion, this second argument is pobably not a valid one.

In Standardised Irish, yes.
But the situation on the ground in the Gaeltacht might not be so clear. After all, Gaeltacht natives are thought to say "obair baile" for homework.
I argue that where baile is an indefinite noun "a townland", then "obair bhaile" would be right for "work of a townland". But where baile is a genericised version of "an baile = home", then there is no lenition, and the Gaeltacht speakers have it right.
So this means that "oifig bhaile" is right, as baile is an indefinite noun, but you see the situation is involved, and this may be why a simple rule was imposed in the CO in order to simplify for learners, rather than necessarily being what the majority of native speakers said.

I think maybe Labhrás has more to say on this, as I may be confusing the picture even more and getting it wrong?

In Ua Laoghaire's works, we have obair snáthaide once, and obair shnáthaide twice for "needle-work".

Cathaoir shúgáin - a straw-rope chair has lenition. This is mentioned as "genitive where the noun refers to what something is made of" in GGBC.

To scold someone: aghaidh bhéil a thabhairt air
But Cnósach Focal ó Bhaile Bhúirne states that some speakers had aghaidh BÉIL.

Aimsir dínnéir - no lenition
aimsir fómhair - harvest-time, no lenition

The nouns in the genitive here are basically generics and not indefinite.

From Ua Laoghaire's Niamh, which I have just republished:

I altered the spelling in my edition, but left "aghaidh mná" without lenition.

There is sometimes a contrast between duine and dhuine in the genitive. Dhuine is indefinite. Duine is generic.
In Amos 6:10, Ua Laoghaire's translation of the Old Testament, there is this:

Gaol dhuine: a person's relative

Michah/Michas 7:6:

Namhaid dhuine: a person's enemy

Compare: namhaid duine: enemy of the human race. You could refer to Satan the Devil in this way.

Lámh duine: this is always without lenition, maybe because it means "a hand such as would be of a human form" (lámh i bhfuirm mar atá ag an nduine). I would be interested if anyone ever found lámh dhuine attested. If so, it would mean "a person's hand", bringing out the indefinite quality of the qualifying noun.

But I think the frequency with which you just mean "a human hand" would mean you would always say that, even if you meant the meaning to be indefinite. But then I came across this in Béaloideas in 1927:

Lámh Dhuine Mhairbh
Béaloideas
Iml. 1, Uimh 1 (Jun., 1927), p. 48 (1 page)
Published By: An Cumann Le Béaloideas Éireann/Folklore of Ireland Society
This is a very short piece, only a couple of sentences from Seán Ó Sé from Baile an Sceilg, Co. Kerry with the phrase lámh dhuine mhairbh in it. Whether the fact that duine itself was qualified by an adjective played any role in that is worth considering.

Unfortunately not.
Lenition rules are very confusing

I bit the bullet and read the Caighdeán Oifigiúil from sections 10.2.6 to 10.2.11. These are the sections concerned with lenition of a qualifiying noun that comes directly the noun it's qualifying. There are really only three or four situations where lenition is called for here. The problem is that there are a load of diverse exceptions. Being in command of all those exceptions while being able to speak without hesitation would be quite a challenge, although I'm sure it is do-able with enough practice.

Djwebb, I wasn't aware of the distinction between indefinite nouns and generic nouns when it comes to lenition but that's good to know.

Per the CO, in turns out that my example of "oifig bhaile" is actually wrong. It should be "oifig baile", despite "oifig" being feminine. The reason is section 10.2.7.j. This lists a number of random nouns inc. "oifig", which, when any of them is acting as the governing noun, lenition does not occur to the qualifying noun.

Interestingly the CO has "obair bhaile".

Section 10.2.7.e has "Ní shéimhítear an dara hainmfhocal más (i) duine, (ii) ainmhí, (iii) dúil bheo, nó (iv) cuid de dhuine,
d’ainmhí nó de dhúil bheo, an dara hainmfhocal". This would explain "aghaidh mná", and in fact it's explicitly given as an example.

All in all, it's fairly heavy going. If you just stuck to the 3 or 4 basic rules, I'm guessing that you'd probably be right >80% of the time.