We can break glass ceiling says Aiden.
The Republic of Ireland international has no doubt that the Blues can go one better than last term and finish in the top four to qualify for the Champions League.
McGeady, who joined Everton in January, has seen enough in the squad to say that Everton are good enough to win a place at Europe's top table.
"We were challenging the top four, so the challenge next season is to go one better and get into that top four," said the winger.
"With the squad we have got, the manager and the backroom staff, the potential that the players have got, especially seeing as how many of the players are 22, 23, 24, young players, that's got to be our aim.
"We probably out-performed what everybody expected of us from the start of the season."
McGeady added: "At the start of the season, the first three games were draws and everyone was thinking, 'This is a different type of football, a different way of playing for Everton'.
"But gradually over time, the manager and the way we play, we have won everyone over.
"We play football the right way and I think everyone is enjoying it."
Blues winger Aiden McGeady