Please Sir, can I have some MORE?

There are two words (at least) that can be use for "more". Firstly if you are trying to convey an increase, say in size, the word πίο/ΠΙΟ can be used. So if you want a larger table than the one being shown to you:
θέλω ενα πίο μεγάλο τραπέζι/ΘΕΛΩ ΕΝΑ ΠΙΟ ΜΕΓΑΛΟ ΤΡΑΠΕΖΙ

or if it's difficult understanding someone you may ask them to speak "more slowly, please"
πίο αργά, παρακαλώ/ΠΙΟ ΑΡΓΑ, ΠΑΡΑΚΑΛΩ

But if you are using more to mean "another" as in an incremental "more", for something that can be counted, then άλλος/ΑΛΛΟΣ is more appropriate. So if you want another beer:
θέλω άλλη μία μπίρα/ΘΕΛΩ ΑΛΛΗ ΜΙΑ ΜΠΙΡΑ

Remember this is an adjective to gender and number mean the endings change. So if it's two beers you are after:
θέλω άλλες δύο μπίρες/ΘΕΛΩ ΑΛΛΕΣ ΔΥΟ ΜΠΙΡΕΣ