I think you can get a rough idea by grouping affixes according to their rough function and assuming that people wouldn't tend to form words with two affixes of a similar function (e.g. it would be rare to make a word with both "bi-" and "tri-", because they both denote conflicting numbers).
Crystal (Cambridge Encyclopedia of the English Language) suggests the following categories for what he sees as the 57 "common prefixes in English" (without giving exact criteria):
- negation ("dis-", "in-")
- reversal ("de-")
- disparaging ("mal-")
- size/degree ("arch-", "sub-")
- orientation ("anti-", "contra-")
- location/distance ("extra-", "intra-")
- time/order ("ex-", "pre-")
- number ("bi-", "uni-")
- grammatical conversion
So as a rough guess, we could say that the upper limit on prefixes was roughly one prefix from each of these categories.
As far as suffixes, adapting also from a list of common ones suggested by Crystal, we might suggest that it would be rare to have more than one from each of the following categories in a single word:
- abstract noun markers ("-dom", "-hood")
- concrete noun/"agent" markets ("-ster", "-eer", "-ist")
- word category markers ("-ly", "-ate", "-ify")
- noun derivation from verb/adjective ("-age", "-ity")
- inflection/adjective derivation from noun/verb ("-less", "-able")
It's more difficult to combine suffixes because of their tendency to change the word class. As a rough guage of the limit on suffixation, maybe we could say it is around 4 (the first three of these categories, plus one instance of word category "derivation", although occasionally you will get 2 of the latter combined).
So as an "absolute upper limit that would apply in 99% of cases", a sensible conjecture on the basis of the above would be around 13.
On the other hand, if you take a word such as "pseudo/anti/dis/establish/ment/arian/ist/ic/ally" with 8 (or 9 if you include the automatically necessary "al") affixes, this appears intuitively to be reaching the upper end of what is practical...
Update: I should also concur that theoretically you can find corner cases where there is no upper limit. Certain prefixes can themselves be repeated. If you can "re-do" something, you can also "re-re-do" it etc. If I'm an anti-abortionist, somebody who doesn't agree with me is an anti-anti-abortionist, and somebody disagreeing with their philosophy is an anti-anti-anti-abortionist. In music, there is an interesting pattern of nomenclature for short notes ("semiquaver", "demisemiquaver", "hemidemisemiquaver", "semihemidemisemiquaver"...). Suffixes which don't change the word category of the derived word (or combinations which change it back and forth) are also potential candidates. "Loneliness" is the concept of being lonely; "Lonelinessless" is a lack of loneliness; "Lonelinesslessness" is the concept of there being a lack of loneliness etc. The nomenclature of chemicals involves a whole system of affixes that can theoretically be used to name compounds of infinite complexity using "words" of infinite length.