As Andrew says, not all the verbs constructed with personal pronouns are true reflexive verbs.
Accorgersi is a so-called PRONOMINAL verb. Pronominal verbs are verbs that do have a reflexive-like form but have no reflexive meaning and don't exist in the non-reflexive form.
For a reflexive verb like lavarsi (to wash oneself), you have the basic form lavare (to wash). On the contrary, accorgersi, or pentirsi (to repent) don't have a pronoun-less (you must always say "MI accorgo" and "MI pento", *accorgo and *pento are incorrect).
So, "accorgersi" is the infinitive form meaning "to notice". The conjugation is the same as for reflexive verbs: io mi accorgo, tu ti accorgi...
It isn't true that "accorgermi" doesn't exist, though. This form with the flected pronoun after the infinitive is used in subordinate clauses when the subject is the same:
- "Ho camminato a lungo prima di accorgerMI che avevo sbagliato strada" ("I walked for a long time before noticing that I had taken the wrong way")
- "Pensa bene a quello che fai, perché poi potresti pentirtene" ("Think thoroughly about what you are doing, because you may regret it later"; pentirti + ne = pentirTENE, "ne" changes "mi", "ti", "si", "ci", "vi" into "me", "te", "se", "ce", "ve": pentirmene, pentirtene...)
Cases like "mangiarsi un bel piatto di pasta" aren't reflexive, either. In the Italian grammar, we call this pronoun an "ethic dative" (dative is a Latin case that shows, i.a., the person to whom you give something, or who takes advantage from something). As Andrew said, they add a meaning of advantage or enjoyment.
If you have more questions about Italian grammar, feel free to get in touch with me.