When "today's" is taken to be a possessive rather than a contraction of "today is", the construction "My today's breakfast" is not grammatical.
It is possible to interpret it as meaning the same as "The breakfast I eat today", but this interpretation is only possible after a pragmatic re-interpretation.
"today's" is not occurring in this construction as an adjective.
In fact, "today's" is never an adjective. Adding the possessive to a noun (or, more accurately, a noun phrase) turns it into a determiner. Determiners are words like articles ('the' and 'a'), quantifiers (for example, 'all' and 'some'), possessives (for example, 'my' and 'your', sometimes misleadingly called possessive adjectives) and demonstratives (for example, 'this' and 'those') and they introduce or determine nouns.
Thus "today's" is a determiner, not an adjective.
All of the following are examples of determiners coming before the word "breakfast":
- The breakfast
- Some breakfast
- My breakfast
- John's breakfast
- Today's breakfast
Adjectives can come between a determiner and a noun, as evinced by
- The tasty breakfast
- Some tasty breakfast
- My tasty breakfast
- John's tasty breakfast
- Today's tasty breakfast
But only certain very special combinations of determiner's can occur together (for example "all" + "that" becomes "all that"). For the most part, determiners never occur next to one another, as evinced by the markedness of
- *The a breakfast
- *Some this breakfast
- *John's your breakfast
- *Some today's breakfast
This is why your example of "My today's breakfast" sounds wrong. Because both "my" and "today's" are determiners. And most determiners don't play well together.
If you just want to know what can come between a possessive determiner like "my" and a noun like "chair," the answer is any number of adjectives, even a list. For example, "My big, wooden, colorful chair."