The example you point to is not analyzing
tabe-sase-rare-ta. It is analyzing
tabe-sase-rare-takatta. In traditional grammar,
takatta is considered the past form of the adjectival affix
tai 'want', and the link you point to analyzes it as such. But if you look at it in more detail, it is actually the contracted form of:
(1) tabe-sase-rare-ta-ku at-ta
`wanted to be forced to eat'
In actual usage, you will not see the string (1) as is because contraction is obligatory. However, once contraction is blocked by some phrase, you will notice that the structure seen in (1) is actually there:
(2) tabe-sase-rare-ta-ku sae/mo at-ta
In Japanese, the affixational morphological slots on verbs, if any, are not as elaborated as you see in Chuckchee mentioned in the link. All you can see in (2) is that particular morphemes select particular category to attach to.
tabe- A verbal stem
-sase- A verbal affix to a verb
-rare- A verbal affix to a verb
-ta- An adjectival affixal root to a verb
-ku An infinitival affix to an adjective root
ar- A verbal stem selecting an infinitive (Turns into at- by gemination)
-ta The past tense affix to a verb
From these selectional properties, the ordering of these morphemes is pretty much limited. Further notice that, in this case, the following alternative ordering, if possible at all, will have a different meaning.
`wanted to be forced into being eaten'
In this case, syntax is playing an important role.
The general topic you are asking is much more complicated than what can be explained in a webpage or two.