I have never heard 'chiquititita', although I have heard people using suffixes in the way you describe.
The root of that word would be chiquito/ta, so you could form 'chiquitito' (no extra 'ti'). As you can see in the definition the word itself is the diminutive form of chico (which can be understood also as an adjetive meaning "small")
With "chiquitito" you would be adding a diminutive to a word that already depicts something small or tiny, but you are not really adding two sets of diminutives to "chico".
You could use other suffixes to enhance the meaning of other suffix.
pequeñitísimo (pequeño -> pequeñito -> pequeñitísimo)
Nevertheless, I have heard "pequeñitito", which I guess is formed in the way you describe
pequeñitito (pequeño -> pequeñito -> pequeñitito or pequeñitico)
so I guess such combination may be possible, although not with all suffixes. Also probably this usage may be understood as colloquial and not as something that adheres to the best practices of the grammar (favor
pequeñísimo over "pequeñitísimo").
For the same same reason, using augmentative and diminutive suffixes for the same word may be possible in colloquial Spanish. Just imagine that you see the baby of a friend of yours, and this baby (chiquito) has grown a lot since the last time you saw him, and you want to convey that the pequeñín is a "big small kid":
Qué guapo está este chiquitote!
Or if your friend had kittens (gatitos) and one was much bigger than the rest:
Eso no es un gatito, es un gatitote!!
Again, this usage is completely colloquial, and the context may be carrying a lot of information that helps to think such words may be just fine to express the idea.