Question: Following a previous discussion about **I would like** vs **I would have liked**, I have a similar question.

What is the difference between

> 
> I wanted to come to your party but I couldn't 
> 
> 
> I would have liked to come to your party but I couldn't.
> 
> 
>

Answer: 
There isn't a rule that you can't use two possessives, but they don't indicate possession of the noun at the end, but instead each one modifies the next phrase.

> 
> Our last week's meeting
> 
> 
> 

Is naturally read as 

> 
> (Our last week)'s meeting
> 
> 
> 

So, unless you are talking about meeting someone with in the week before you both die, it is unlikely to mean what you intend.

Multiple possessives tend to combine left to right, so 

> 
> your mother's old uncle's shepherds pie
> 
> 
> 

Is the pie of a type associated with shepherds belonging to the old uncle of your mother. 

> 
> (((your mother)'s old uncle)'s (shepherds pie)
> 
> 
> 

It doesn't mean that there is an old pie which belongs to you, your mother, an uncle and a number of shepherds all at the same time!

The example of course proves the rule - 'shepherds pie' is an idiomatic noun phrase so doesn't combine.
You can use two possessives in a row, that is not an issue. For example, if I have a cat, and my cat has a toy, I can say "my cat's toy." If there is a manufacturer for that toy, it would be somewhat acceptable, although convoluted, to say "my cat's toy's manufacturer," which would be three possessives in a row. If that manufacturer had subsidiaries, it would be much harder to follow, but you could probably still say "my cat's toy's manufacturer's subsidiaries." Though, with four possessives in a row, we'd probably rearrange the sentence structure to break them up some. 

In English, we would commonly say "last week's meeting," or perhaps "our meeting last week." We don't put an "our" modifier before a chronological modifier, such as "our last week's meeting," or "our today's meeting." The expectation is that "our" probably modifies the word that directly follows it, in other words, it sounds like "our" modifies "last week." But "last week," doesn't belong to us, so this sounds very weird. Unless you actually need the "our" modifier to clarify whose meeting *and* you need the "last week" modifier to clarify the time of the meeting, it makes sense to drop the modifier that is less helpful at conveying meeting.