1 syllable:
aim, aimbe, aime, ame, baim, bame, blame, bleyme, boehm, boehme, brame, came, chame, claim, crame, dame, damme, fame, fayme, fe mm, flaim, flame, fraim, frame, game, graeme, grame, haim, kaim, lame, maamme, maim, mame, mayme, payme, prame, rhame, rtgame, same, scame, sejm, shame, sname, swaim, tame, trame

2 syllables:
a-frame, aa mm, acclaim, aflame, agame, agname, airframe, arkaim, ashame, ballgame, ball game, became, bedframe, beldame, big-name, big game, birthname, boardgame, board game, brand-name, brandname, by-name, byname, card game, chess game, child's game, code-name, coldframe, cold frame, con game, court game, crap game, cross-claim, dagame, day game, declaim, defame, diblaim, diffame, disblame, disclaim, disclame, disfame, doorframe, dulaim, efraim, eglaim, ename, enflame, enframe, entame, essaim, exclaim, fair game, fangame, field game, filename, first-name, foregame, freeze-frame, gameframe, golf game, goodgame, go game, grandame, grande dame, hangame, headframe, head game, hogframe, home game, hutaym, icame, iframe, igname, ill fame, iname, inflame, keyframe, khadgam, lay claim, lovegame, mainframe, make-game, matched game, match game, mazame, misclaim, misframe, misname, mizraim, moorgame, nachname, nandaime, nickname, night game, no-name, noname, nonclaim, nongame, ogame, pathname, pay claim, pen-name, philem, place-name, playgame, postgame, pregame, prename, proclaim, pygame, quiname, quitclaim, r.a.m, rakeshame, re-claim, reclaim, reflame, reframe, reklame, rename, rephaim, road game, schooldame, self-same, selfsame, shell game, sirname, soname, spaceclaim, sports game, stealth game, stepdame, sub-frame, subframe, subgame, surname, take aim, tannaim, telaim, the same, timeframe, to-name, two-name, uname, unframe, vidame, vorname, wage claim, war-game, warframe, wargame, war game, wegame, wireframe, word game, ysame

3 syllables:
abel-maim, acesulfame, action game, adoraim, advergame, aftergame, all the same, amoraim, apodeme, arcade game, aspartame, away game, banking game, baseball game, basename, bloody shame, browser game, burlingame, call-by-name, candle flame, cname, codename, counter-claim, counterclaim, cover-shame, darwin's game, dateiname, disinflame, doppelname, dragonframe, edamame, ekename, ender's game, eurogame, fireflame, first they came ..., forename, gacha game, guessing game, hall of fame, hippodame, hockey game, horonaim, indie game, initgame, khaby lame, kirjathaim, lanciname, laying claim, megagame, middlegame, minigame, misbecame, molly's game, neotame, numbers game, open frame, oriflame, outdoor game, overcame, overname, parlor game, party game, passing game, perfect game, picture frame, pinball game, placename, playoff game, poker game, practice game, racing game, running game, samegame, sayhername, sense of shame, sepharvaim, set aflame, sitename, softball game, spinning frame, stickball game, superframe, table game, tokoname, tradename, truename, underframe, username, waiting game, whatshisname, whatsisname, window frame, without aim

4 syllables:
abel-mizraim, amerigame, athletic game, basketball game, beheading game, body-on-frame, chinese surname, computer game, confidence game, dinosaur game, gambling game, inertial frame, insurance claim, one and the same, reference frame, role-playing game, survival game, the grand tour game, videogame, video game, volleyball game

5 syllables:
aram-naharaim, battle royale game, chushan-rishathaim, embroidery frame, exhibition game, sports video game, wansapanataym

6 syllables:
action-adventure game, action role-playing game, hollywood walk of fame, new york mets hall of fame, puzzle video game, racing video game, rock and roll hall of fame, the imitation game, wwe hall of fame

7 syllables:
inertial reference frame, pathfinder roleplaying game, role-playing video game, strategy video game, the hunchback of notre-dame

8 syllables:
multi-player video game, multiplayer video game, simulation video game, single-player video game