[4] After moving to the academy of FC Nordsjælland in 2010 from childhood club Brøndby IF, Maxsø made his professional debut in 2012 against Copenhagen.

[7] In his early days at Brøndby, Maxsø did not stand out as a player as he was neither big nor particularly quick, but those who worked with him noted his constant desire to improve various aspects of his game.

[7] In 2010, he joined the FC Nordsjælland youth academy and had a large growth spurt that made him taller and physically stronger, which meant that he could transition to defense more easily.

[9] Maxsø signed a contract extension with Nordsjælland in the summer of 2016 and turned down a transfer offer from Danish top side FC Copenhagen, citing a desire to play abroad.

On 11 January 2017, Maxsø revealed his intentions of leaving Nordsjælland in the summer of 2017 with the goal of being sold after the 2017 UEFA European Under-21 Championship.

[13] He would struggle to find his footing in Osmanlıspor, and was taken out at halftime in the 4–2 loss to Sivasspor on 17 September after being a part of a defense conceding four goals before half.

[20] On 25 October, he was part of the starting lineup as Zürich won 3–2 over Bayer 04 Leverkusen in the Europa League group stage.

[23] He finished the season with 36 appearances for the club in which he scored one goal,[15] as Zürich ended seventh in the league missing out on European qualification by two points.

[27][28] However, as the transfer window closed, Maxsø explained that he was 'stuck' in Uerdingen 05 and had decided to buy himself out of his contract; something which according to him cost him millions of Danish kroner, but was a necessary career move.

[31] On 25 September, Maxsø scored his first goal for Brøndby, a vital equalizer in extra time in the Danish Cup against Skive IK.

[37] On 11 December 2020, Maxsø received the award of Brøndby Player of the Year 2020, beating out fellow nominees Anthony Jung and Morten Frendrup.

[43][44] Maxsø would instead stay at Brøndby, and made his European debut for the club on 17 August in a 2–1 away loss to Red Bull Salzburg in the first leg of the play-off round of the UEFA Champions League.

[46] On 27 January 2023, Maxsø signed with Major League Soccer side Colorado Rapids on a three-year deal with a club-held option for the 2026 season.