Translation:I am washing the dishes.
Perhaps not a compound, but it is a derived word.
mos is "to wash", and -ogat is apparently a form of the suffix -gat/get which is used to derive frequentative forms (as beszél "talk", beszélget "chat, converse").
So it was originally "wash again and again" or something like that, but then took on the specialised meaning of "do the dishes".
(German also has a single word, incidentally: abwaschen. Compare English "do the washing up", which is similar but needs a light verb "do".)
Maybe the frequentative is because you need to wash several pieces? I do know know. But yes, it is a frequentative and it means doing the dishes. It is -o-gat, I would say, the "o" being a buffer sound.
Mosni - to wash - without the frequentative, means to do the laundry.
And if you wonder how you would wash yourself, that is mos-a-kod-ni. The "-kod" makes it, what is it, reflexive(?).
So, for these three - mosni, mosogatni, mosakodni - you don't need to add anything else, they by themselves mean doing the laundry, the dishes and washing yourself (respectively, of course).
For anything else, you use "mosni" plus the object being washed (car, hair, windows, hands, whatever).
Oh, and there are other specialized frequentatives. The word "látogat" comes to mind.
lát - see
lát-o-gat - visit
It does not need to be frequent. Even one visit is a "látogatás". ("-ás") makes it a noun. Hungarian is so much fun.
Technically, mosok = I wash/am washing whereas adding the -gat suffix implies a repetitive or recurring action (think beszélek = I talk, beszélgetek = I have a conversation).
So even by a literal interpretation, mosogatok is more like a process of washing a bunch of things than just the action of washing.
However, the verb mosogatni does not mean washing just any set of things, like washing the laundry, it means washing dishes. I don't think you can derive this meaning from the root and suffixes, it has simply gained this specific meaning.
To illustrate further, a mosogatógép (lit. "mosogat-ing machine") is a dishwasher whereas a washing machine (for clothes) is a mosógép (lit. "mos-ing machine"). You can look up mosogatógép and mosógép on images.google.hu to see for yourself! :)
"I wash the dishes" = "Én mosogatok" "I am washing the dishes" = "Én mosogatok" If you need to state that you are doing it right now, you would say... "Én most mosogatok". Hungarian doesn't have the present-continuous tense that English does. Hungarian only has 3 tenses, present, past and future. No present-perfect, etc., Just the 3.
"I am the dishwasher" = Mosogató vagyok.