If either party (i.e. Democrats or Republicans) in the US would proceed to "pack the Supreme Court", what would be stopping the next administration (i.e. after power has switched once again to the other party in an election) from just doubling (plus one) the number of judges again, in order to re-gain a majority of judges that are to their own liking? Nothing?
Functionally, nothing except the regular requirements of passing legislation and nominating justices.
Altering the number of judges on the Supreme Court is as simple as passing legislation through the normal channels, as the Constitution is silent on the number of judges on the Court. This was last altered in 1869, when the Judiciary Act of that year set the number at nine; the Chief Justice, and eight associate Justices.
Assuming a presidential administration has the numbers in the House and the Senate to pass such legislation it follows that they also have the required numbers to confirm a nominee, as the elimination of the filibuster for Supreme Court nominations in 2017 ensures that a party in control of the Senate may pass a cloture vote by simple majority in order to end attempts to delay a confirmation vote.
Attempts to filibuster the legislation to expand the Court could be successful, depending on the partisan composition of Congress, however given the precedent set in 2017, it is perfectly possible that a filibuster could be overridden by another use of the 'nuclear option'.
Politically, however, the consequences for the credibility of the Court and the Judiciary as a whole would of course be significant, which could make an administration pause before retaliating. The danger of inviting a tit-for-tat response was, in fact, acknowledged by Joe Biden during the fourth Democratic debate in October 2019:
Erin Burnett: Vice President Biden, the Constitution does not specify the number of justices that serve on the Supreme Court. If Roe v. Wade is overturned on your watch and you can't pass legislation in Congress, would you seek to add justices to the Supreme Court to protect women's reproductive rights?
Joe Biden: I would not get into court packing. We [add] three justices. Next time around, we lose control, they add three justices. We begin to lose any credibility the court has at all.
In theory, nothing prevents this. In practice, though, it's not a decision that lies with the president, so it requires a majority vote of both houses of congress in addition to the president's signature. It's therefore unlikely to occur every four years or eight years even if the present reluctance to increasing the size of the court might abate. Furthermore, at some point the justices themselves will probably start objecting if the court's size becomes unwieldy, thereby making it difficult to conduct the court's business.