I should have went

OK, stop shuddering now. This particular construction seems to be high on the hackle-raising scale. "Illiterate!!!" people will thunder.
Because, well, the conjugation of the verb "go" is so logical. I mean, "Go, went, gone", what's not intuitive about that??
Why IS the word "go" so weird?
And why do so many people use "went" as both the past participle and the simple past, though "have went" is so OBVIOUSLY WRONG!!!!
Back in Anglo-Saxon times, "go" was fairly easy to conjugate. The infinitive was "go", the past participle was "gone" and the simple past was.... "goed".  Bet you didn't see that one coming. Young children still conjugate it this way.
There was another verb, "wend", which meant "move, turn, or change direction", and gradually came to mean "go in a certain direction". This survives now only in the phrase "wend one's way". The simple past was "wende". But from about 1200 on, the form "went" started to be used for both the simple past and the past participle of "wend": I wend, I went, I have went. Because of the similarity in meaning between "go" and "move in a certain direction", "went" migrated over to the verb "to go" and settled in there by about 1500 as its simple past, booting out "goed", and, in some varieties of the language, "gone" as well. Here's the proof:

1642   W. Sedgwicke Zions Deliv. Ded. sig. A2^v,   A Judge that would have went right if the times had not beene bad.

1729   S. Switzer Hydrost. & Hydraul. 319   The Length of Time it [an engine] has went.

Clearly someone who is writing about hydrostatics and hydraulics is not illiterate!

But, for whatever reason (there are never logical reasons in the language), by the 19th century, "went" was stigmatized as the past participle of "go", though considered perfectly correct for the simple past. However, the past participle "went" is obviously still alive and well, since many people still say "have went".

There is always a historical explanation behind language usage.  I'm not suggesting that you should use "have went", because it really is stigmatized, but those of you who think it's the most terrible solecism, please do recognize that it COULD just as easily be the standard past participle of "go" as "gone" is. People who say "have went" are just the linguistic heirs of that 18th-century guy writing about hydrostatics. If their version had been the prestige dialect, the "have went" people would now be looking down their noses at those who say "have gone".

 As for "wend", because its past tense had been shanghaied by "go", it became a regular verb: wend, wended, wended.