One noun but two determiners?
Solution 1:
[Note: Here, as elsewhere on EL&U, I try to use capitalised words for grammatical relations/syntactic functions and small case only for terms indicating word categories and phrasal categories.]
Preamble
Traditional grammars, as well as modern grammars such as Oxford Modern English Grammar (Aarts 2011) or The Cambridge Grammar of the English Language (Huddleston & Pullum 2002), recognise that English noun phrases usually have two main parts. So with regard to the phrase those pink elephants, the phrase comes in two chunks. The first is the word those and the second is the string pink elephants. This second chunk is of course itself a phrase, not a single word. In order to distinguish phrases like pink elephant from the larger noun phrases they occur in, they are referred to here as nominals.
The Original Poster's Question
According to frameworks like the one used in The Cambridge Grammar of the English Language (Huddleston and Pullum, 2002), there is only one Determiner per noun phrase. Within CamGEL, the term Determiner is short for the more explicit label Central Determiner. This is necessary because such grammars also recognise a separate function Predeterminer. So in the phrase:
- All the elephants
the word all is in Predeterminer function and the word the is in Determiner function. Notice that in terms of word category, both the and all are determinatives. The word all, like all other determinatives can also occur in Determiner function, as in the phrase:
- All elephants
The fact that we can only have one Central Determiner per noun phrase is born out by the ungrammaticality of phrases such as :
- *the my friend
- *some the people
- *any his words
- *no those friends
Cardinal numbers, which CamGEL regard as determinatives, can occur within this framework both as Determiners, or as Modifiers within nominals:
- Three elephants
- The three elephants
In example (2), the word three is occurring in Determiner function and the nominal, the Head of the noun phrase, is the item elephants. In example (3), however, the Determiner is the word the and the nominal is the phrase three elephants. Within this nominal the word three is a Modifier and the word elephants is the Head.
In the original Poster's example:
- Another three days
The determinative three is occurring as a Modifier within the nominal three days. The determinative another is occurring in Determiner function within the larger noun phrase another three days—according to the CamGEL analysis.