Here is how you derive the past tense for 行く (and the te-form, equivalently):
past tense of /iku/
== { past tense of X = continuative form of X + /ta/ }
/iki/ + /ta/
== { generating a new word requires sound-changes }
repair(devoicing(/ikita/))
== { /i/ and /u/ become devoiced/disappear in-between two devoiced consonants }
repair(/ikta/)
== { normally, I-replacement repairs /k/+consonant; but here, gemination }
/itta/
Basically, it comes down to a devoiced vowel resulting in a consonant-consonant sequence, and that CC
sequence getting repaired somehow.
For all other -ku verbs, e.g., /kaku/
, we get /kakita/ => /kakta/ => /kaita/
, using I-replacement.
So the exception with /iku/
is that gemination repairs the /k/+C
sequence instead of I-replacement.
That is the only exception I know which pertains to repairs, to directly answer your question (instead of the question "what words conjugate weird?").
As to why this happens with 行く, I am not sure... in 徒然草 (1330-1332),
用有りて行きたりとも、其の事 果てなば、とく歸るべし
and even much more recently, in 不如帰 (~1898)
今まで知らぬ自由と楽しさのこのさきに待つとし思えば、父に別るる哀しさもいささか慰めらるる心地して、いそいそとして行きたるなり。
we indeed have proof it was, at least in writing, the continuative form plus the past(/perfect) morpheme, with no repairs. At some point, this changes, see こゝろ (1914):
私は早速先生のうちへ金を返しに行つた。
so this doesn't really help us figure out anything very much at all. I can't seem to track down anything useful here.
I'd classify 問う and 請う as a different "exception" from 行く, because for whatever reason, a totally different sound change happened, from -ひた- to -ふた-. We can still see the older form here,
女に思ふやと問ひたりけれどいらへもせざりければ
思ふともいはずなりぬる時よりも増る方にて頼まるゝ哉
from 左近大將朝光 in the 後拾遺和歌集 (~1086). This -うた form is still around in the Osaka dialect and is used for the past tense and te-form regularly. It's not clear to me why there was this branch in pronunciation.