#epithets

Areios (Ἄρειος), “warlike.” 

Arretos (Άρρητος), “ineffable, unutterable.”

Bakkhos (Βακχος), “frenzied.”

Bassareus (Bασσαρεύς), “the fox.”

Boukeros (Bουκερως), “bull-horned.”

Bromios (Βρόμιος), “the roaring.”

Botrokhaites (Bοτρυοχαίτης), “with leaves in his hair.”

Dendrites (Δενδρίτης), “of trees.”

Dikeros (Δικέρως), “two-horned.”

Digonos (Δίγονος), “twice-born.” 

Dimetor (Διμήτωρ), “of two mothers.”

Dimorphos (Δίμορφος), “two-formed.”

Diogenes (Διογένης), “son of Zeus.”

Diphyes (Δῐφῠής), “dual-natured.”

Dithyrambos (Διθύραμβος), “of the dithyramb.”

Eiraphiotes (Ειραφιωτης), “insewn” [referring to his birth]

Eleutherios (Έλεύθεριος), “the liberator.”

Eubouleos (Εὐβουλεύς), “of good counsel,”

Eukhaites (​​Ευχαίτης), “lovely-haired.” 

Euios (Εὔιος), “howling” [referring to the ecstatic cry “euoi.”]

Eumenes (Εὐμενης), “fair-minded, kind.”

Gelotos (Γελωτος), “of laughter.” 

Gethosunos (Γηθόσυνος), “joyful.”

Gonoeis (Γονόεις), “virile.” 

Himertos (Ίμερτός), “desirable.” 

Hiraphiotes (Ίραφιώτης ), “hierophant”

Hulēeis (Ύλήείς), “forest-dweller.”

Hyes (Ὕης), “of moisture/the dripping.”

Karpios (Καρπιος), “of fruit.”

Keraos (Κεραός), “horned.” 

Kharieis (Χαρίεις) “graceful.” 

Khoreutes (Xορευτές), “the dancer.”

Khthonios (Χθόνιος), “of the Underworld.”

Kissios (Κισσιος), “of the ivy.”

Kissobryos (Κισσόβρυος), “adorned with ivy.”

Kissophoros (Κισσοφορος), “ivy-bearer.” 

Kissostephanos (Κισσοστέφανος), “ivy-crowned.”

Komastes (Κωμαστής), “reveller.”

Kryphios (Κρύφιος), “hidden.”

Lenaios (Λήναιος), “of the wine-press.”

Limnaios (Λιμναίος), “of the marsh.”

Lyaios/Lysios (Λυαῖος/Λύσιος), “loosener/deliverer.”

Lyknites (Λυκνίτης) “winnower/in the cradle.”

Mainoles/Mainomenos (Μαινόλης/Μαινομενος), “the mad.”

Melanaegis (Μελαναιγις), “of the black goat-skin.”

Meilichios (Mειλίχῐος), “the mild/gracious.”

Melpomenos (Μελπομενος), “singer, of the tragedy.”

Methydotes (Μεθυδώτης), “giver of wine.” 

Mystes (Μύστης), “of the Mysteries.”

Nebridopeplos (Νεβριδόπεπλος), “clothed in fawnskin.” 

Nyktelios (Νυκτελιος), “of the night.”

Nyktipolos (Νυκτιπολος), “night-stalker.”

Nyseus (Νυσεύς), “of Nysa.” 

Oineus (Οἰνεύς), “of wine.”

Oinops (Οινοψ), “wine-faced.”

Omadios/Omophagos/Omestes, (Ώμάδιος/Ώμοφάγος/Ωμεστης), “eater of raw flesh, carnivore."

Orthos (Ορθος), “erect, upright.”

Phleon (Φλεων), “the luxuriant.”

Philomeides (Φιλομειδης), “laughter-loving.”

Phriktos (Φρικτός), “scary.” 

Ploutodotes (Πλουτοδότης), “giver of riches.”

Polygethes (Πολυγεθής), “of many joys.”

Polykomos (Πολυκῶμος), “of many revels.”

Polymorphos (Πολυμορφος), “many-formed.” 

Protogonos (Πρωτόγονος), “first-born, primordial.”

Psilas (Ψιλας), “giver of wings.”

Psilax (Ψιλαξ), “uplifted on wings.”

Psykhodaiktes (Ψυχοδαϊκτής), “render of the soul.” 

Soter (Σωτηρ), “savior.”

Staphylites (Σταφυλίτης), “of the grape.”

Tauropos (Ταυρωπός), “bull-faced.”

Teletarkhes (Τελετάρχής), "Lord of Mystery Rites"

Thelymorphos (Θηλύμορφος), “womanish/effeminate.”

Theoineos (Θεοινος), “god of wine.”

Thyonidas (Θυωνίδας), "son of Thyone/raving."

Thyrsaphoros (Θυρσοφόρος), “thyrsus-bearing”

Trigonos (Τρίγονος), “thrice-born.”

Trieterikos (Τριετηρικός), “of the three-year period.”

Zakholos (Ζάχολος),  “very angry”

First we got his infamous epithets that have to do with the word "polys" aka "much" or "many". We know that Odysseus has several epithets with this preposition. They say that out of all the epithets involving the word "poly" Odysseus has around eight attributed to him, some of them exclusive to him or gods like Hermes.:

πολύτροπος -> polytropos= man of many turns/ways (given that τρόπος stands for "way" but it could also be "τροπή" that stands for "turning")

πολύφρων -> polyphron = ingenius, clever, smart, wise coming from πολύς aka "much" and φρήν which stands for "mind" so "of much mind" or "clever" or "ingenius"

πολύμητις -> polymetis = of many wiles, wily, of many councils epithet including the word μήτις which stands for "wisdom" but also "wile" and "council"

πολυμήχανος -> polymechanos = recourceful, of many resources, of many ingenious ideas, inventive, of many devices including the word "μηχανή" which literally means "machine" or "device" so it is someone that invents ideas or anything similar

πολύτλας -> polytlas = much tormented, of many torments including the verb τλάω which means "to endure" or "to endure torments"

πολύαινος -> polyaenos = much-praised, much-talked about involving the verb αἰνέω which stands for "tell tales" or "speak of" but in this context it has double meaning but mostly positive

πολυάρητος -> polyaretos = much prayed for called such by Euryclaea from the word αράομαι which stands for "pray" or "wish" but also ironically "to pray against" or "to curse"

Of course the infamous epithets for sacking cities or others that have to do with his torments, his skills or his heritage:

πτολίπορθος -> ptoliporthos = sacker of cities standing from the word πτολίεθρον aka "citadel" or "city with walls" and the verb "πέρθω" which stands for "to take" or "to conquer", "to sack" or "to destroy"

τλήμων -> tlemon = tormented, steadfast, enduring from the verb "τλάω" which means "to endure" or "to be tormented"

λαερτιάδης -> laertiades = son of Laertes

δουρικλειτός -> douriklaetos = famous for his spear, famed spearholder involving the words δουρι archaic form of "spear" and κλυτός aka "famed" or "known for"

ποικιλομήτης -> poekilometis = of various wiles including the word ποικίλος which stands for "with variety" and μήτις "wile" or "wisdom" or "plan"

ὠμογέρων -> omogeron = of early old age, green old man, active old man not really an epithet, more like a description but I thought I should include. Deriving from ὠμός which means "raw" or "uncooked" or in this case "not yet fully entering" and γέρων aka "old man" so "someone who has yet to enter old age" or "barely entering old age"

ἐσθλός -> esthlos = noble, good, hapless

ἀμύμων -> amymon = hapless, blameless, good

κυδάλιμος -> kydalimos = glorious

φαίδιμος -> phaedimos = radiant, shining

Ἰθακήσιος -> ithakesios = of Ithaca, Ithacan

ἄναξ -> wanax = king, lord, ruler

δύστηνος -> dystinos = poor, unfortunate, unhappy

ὄλβιος -> olvios = happy, blessed

Then we have epithets that have to do with the word "divine" aka "dios" or "theos". The epithets of that kind are common in greek heroes because not only are they considered similar to gods but also have their line connected to gods:

δῖος -> dios = divine, godlike epithet pretty common for heroes in general given that most of heroes have some sort of relation to gods or goddesses one way or another (it might also be freely translated as "brilliant" or "sublime")

διογενής -> diogenes = born from gods, sprung of Zeus including the verb γίγνομαι which means "to be born" in this context

ἀντίθεος -> antitheos = equal to gods standing by the preposition "ἀντί" which stands for "against" or in some cases "touching against" or "instead of" so in this case "touching gods" or "substitute of gods" aka "equal to gods"

θεῖος -> theios = divine

Then we have the epithets that have to do with his heart and mind and usually connected to the word "phren" which stands for "mind" or "heart" as well as the word "hetor" which stands for "heart":

δαίφρων -> daiphron = warrior, war-like, the one who thinks of war coming from the words δάϊς which stands for "war" and φρήν aka "mind" so stands for someone who has a mind for war so sometimes I have seen it as "war-like" or simply as "warrior". Also common for many heroes

ταλασίφρων -> talasiphron = of steadfast heart, tormented in mind, of patient mind from the verb τλάω which means "to suffer" or "to endure"

μεγαλήτωρ -> megaletor = big-hearted, great-hearted including the word "μέγας" aka "big" and ήτωρ which stands for "heart"

μεγάθυμος -> megathymos = big-hearted, great-hearted, great-spirited same first word as above but instead we have the word θυμός aka "heart" or "soul"

Now others that might include more words:

Διὶ μῆτιν ἀτάλαντος -> dii metin atalantos = equal to Zeus in council

Διῒ φίλος -> dii philos = dear to Zeus

Τηλεμάχοιο φίλος πατέρας -> telemachoio philos pateras = father to dear Telemachus

φίλος πόσις -> philos posis = dear husband

μέγα κῦδος Ἀχαιῶν -> mega kydos Achaeon = great pride of the Achaeans

Some negative epithets or characteristics coming mostly from lips of enemies:

δόλων ἄατος (ἠδὲ πόνοιο) -> dolon aatos (hede ponoio) = insatiable in deceipt (and work -here: deceitful-) involving the word δόλος aka "wile" or "deceitful plan" and ἄατος "insatiable" or "unsatisfied with". The word πόνος means "work" in general but here probably means the negative essence of plan or deceitful work

ὀλίγος τε καὶ οὐτιδανὸς καὶ ἄκικυς -> oligos te kai outidanos kai akikys = little and puny and insignificant again not really epithet but rather the description Polyphemus gives on Odysseus in his anger

ἐλέγχιστος ζωόντων -> elechistos zoodon = vilest of all that live once more not really an epithet but how Aeolus calls him in his fear once he finds out he is cursed by the gods

θρασύς -> thrasys = bold and reckless, cheeky, with audacity called that by Eurylochus

σχέτλιος -> schetilos = rush, reckless but also unflinching, being able to hold out both of negative and positive connotations. He is called that by Achilles in the underworld

κερδαλέος -> kerdaleos = crafty, wily, who seeks his own gain playfully called so by Athena

ἐπίκλοπος -> epiklopos = tricky, thiefy ibid

Sooooo Sorry I shall stop now! This is already too long and definitely forgot several! But from a small search, mainly to the homeric poems (some mentions from the Aeneid being: inventor = crafty, wily, inventor saevus = fierce, notorious, savage fictor = sculptor but here mostly liar, teller of tales of fiction)