Bear with me, please... I'm going to start on what seems like a somewhat tangential topic, then come back around to your question because I think the first informs the latter.
The first game will likely be "who can disclose the least information while we get to know each other and decide if we can trust each other." The book "The Three-Body Problem" won the Hugo in 2015 for laying out a strong argument that the first sentient species to detect another sentient species should probably launch an all out genocidal attack on the other species without ever disclosing that the attack is coming -- that tends to put a damper on other types of games. The reason for this being the correct strategy is pretty simple: resource consumption. Deeper analysis of this argument is outside the scope of this question, but I find the argument rather compelling. There are many other reasons, explored on this forum, that suggest that we won't be playing games with alien species. EXCEPT...
That particular argument and most of the similar arguments for us being in conflict cease to apply if the two species have 100% no interest in each other's worlds and resources. So a species that colonizes the upper atmosphere of gas giants and needs to stay far away from a sun to be at a comfortable temperature wouldn't be interested in Earth, and vice versa. We might have some conflicts over mining resources in the asteroid belts, but those aren't genocidal arguments.
Which brings us to games.
If two species are sufficiently dissimilar that they aren't interested in each other's homeworlds, then they may get to know each other well enough for games. But if you accept the argument that two such species must be from wildly different worlds, then we aren't looking at anything personally physical. No soccer, baseball, etc. Those would all be ruled out not just by different physiology but from the physical inability to exist in each other's physical locations.
Mechanically physical seems feasible -- think NASCAR in space. Three loops around the sun, first to graze Phobos wins. Similar events where the challenge is some sort of engineering challenge would be viable games.
The intellectual games would come with time, and presumably would be whatever humans are playing at the time. If we have sentient species, we would be building up communication for trade and general knowledge sharing, and eventually communicating tic-tac-toe and checkers and chess and Go -- but they PROBABLY already know Go.
[Tangent: Go is a game played on a grid with black/white stones. It has only 9 rules, and variants of the game have, according to unverifiable legend, arisen independently in various parts of the world in human history. This has given rise to the theory that since this grid-of-stones seems like such an intuitive game, and its strategy is surprisingly deep, that all sentient species would at some point stumble into it. Therefore, aliens already know Go. Go study the rules of Go and decide whether it is something you think could just naturally develop in any species that spends time studying geometry.]
Anyway, after we tire of Go, we could communicate with them any game we want.
Now... on that note... why would we tire of Go? Because it is a total-knowledge game. All total-knowledge games will eventually be solved by computers, and it will simply be who has the deeper read of the game tree. Tic-tac-toe is fully solved... so is checkers... others will come with more computation. Ian Banks, in his novel The Player of Games, posited that only games with some element of chance in the early stages would continue to be interesting, and it was best if that chance element would lead to a game that had never been played before. In his novel, multiple species compete in various games with different handicaps between players of different skill, but always with some random element thrown in. Think like Chess, but where the board is the topology of a randomly chosen continent. Or poker, but with a randomly chosen number of cards (no one would ever see the full deck, so you could never be sure how many cards are in any given suit when bidding). That sort of thing. I recommend reading the novel -- over the course of a few hundred pages, Banks delves deep into various kinds of games and why they work or don't work across cultures, far more than I can summarize here.