I'm really late to the conversation but since it was never settled, I will throw in my understanding.
1. Domain Intitiate never clearly states that you can take it multiple times. Most feats have a special property letting you know that you can. DOMAIN INTITIATE lacks this special property.
2. Advanced Domain never even vaguely suggests that you increase your focus pool with a focus point. It just allows you to use the focus point obtained with Domain Initiate to be used with its corresponding Higher spell. Since the domain spells come in a level 1 and level 4 choice.
Based on this, it appears that you only get 1 focus point as a cleric without multiclassing.
The only caveat I catch to my interpretation, is that Advanced Domain has a special property allowing it to be taken multiple times; but since Domain spells only come with two spells in each domain. You would need to take a second instance of Domain Intitiate to even open up the viability of taking Advanced Domain more than once, since you have to have the lesser form of the domain to act on Advanced Domain, as it states " domains which you have an initial domain spell".
I would assume that Domain Intitate would be taken multiple time despite it missing the words that say so.
I would also assume that since it's a different domain you would in fact get another focus point
However Advanced Domain does not grant another point. It just allows more options with the point you got from Domain Initiate.
In conclusion I feel it would be by 6th level a cleric could max out their focus pool. By taking Domain Initiate 3 times
