Let’s say George Washington retires on schedule after two terms. He goes back home, but instead of dying two years later at age 67, he lives well into his 70s as an elder statesman, unifying the Federalist party. His Vice-President, John Adams, is elected in his place as happened historically. However, Washington's continuing prestige is enough to swing a historically very close re-election fight from Thomas Jefferson to Adams, in spite of the controversial Alien and Sedition Acts.
Adams is very much a Republic kind of guy, but with the Democratic/Republicans (main opposition to Adam's Federalists) out of power for another four years, they become more radically anti-federalist. They were already purporting to annul federal laws in some states. Maybe a radical fringe actually rebels against the Federal government or conspires with Spain to seize some of the western territories, as Aaron Burr has been accused of doing. Historically, John Adams lived another 20+ years after his presidency, so maybe he decides to run for a third term, given the radical nature of the opposition and maybe gets a nod from Washington to go for it. (Not sure if Adams would go for it, and really not sure that Washington would endorse him, but if the opposition was looking more radical, maybe Adams would run and Washington might support him). That part is both crucial and iffy.
So now we have basically the same party in power for the first twenty years of the US constitution, (though Washington was not officially a Federalist). We've had no examples of a peaceful change of power between political parties, and John Adams has a politically active son (who later became the sixth president). John Adams has Washington's nod for a third term. Maybe Washington dies before the question of a fourth Adam's term comes up, but the precedent of more than two terms is set. Adam's son, John Quincy was born in 1767, and didn't historically become president until 1825, so somehow they would have to close the gap between a fourth John Adams term, which would end in 1812, and John Quincy becoming old enough to be considered ready for the presidency. Probably one more John Adams term would do it, taking us to 1816. John Adams retires. John Quincy Adams takes over the family business of being president. Then all it takes is a politically active third generation in the family and the US is a monarchy in everything but name.
BTW: Both Adams family presidents were very committed to Republican ideal (the ideal of a Republic, not the modern party of that name) and would have been appalled at the idea of becoming a dynasty if the issue was put to them that way. They would have to be pushed into becoming a de facto dynasty by political pressures and fear of the opposition if this was going to happen.
True, historically the Federalist party kind of imploded, not too far into the game. On the other hand, the opposition would probably become more radical the longer they were out of power. And that opposition had Aaron Burr among them, which could lead to very destructive in-fighting. With a more extended time out of power, I can very easily see the anti-federalists sprouting a more radical, violent wing that tries to forcibly resist the federal government, and (as noted) maybe even trying to set up its own country in the west (as Burr has been accused of trying to do historically). The anti-federalists would claim that the American Revolution has been hijacked and has essentially recreated the British monarchy, while the Federalists would point to, and genuinely fear, the anarchy of the French Revolution and also fear that the anti-federalists want to bring back the weak central government of the Articles of Confederation.
This could end up as a very different US. Jefferson could get away with the Louisiana purchase because his credentials as an anti-federalist were strong. I'm not at all sure Adams would make the same decision, because he would be more vulnerable to accusations of abuse of power. So probably no Louisianna purchase, which meant a much smaller US, at least for a while. As to who would end up with the Louisianna purchase, I don't know. The British might invade it when they returned to war against Napoleonic France. It might become part of Mexico, though undoubtedly with litle real Mexican control for many decades. Maybe it would become a refuge for anti-federalist settlers carving out mini-republics in the chaos, with nominal loyalty to Spain or Mexico. Those republics would have nearly twenty years to develop under very tenuous Spanish rule before Spain was kicked out of Mexico.
The federalists had a strong anti-immigrant wing, partly a reaction to the excesses of the French Revolution. They would probably clamp down on immigration, slowing US population growth, at least until industrial expansion revealed the need for a bigger workforce.
So, a US that remains smaller and less populated, at least for the first several decades. A chaos of weak backwoods Republics hostile to the US government on the frontier. An increasingly entrenched elected Adam's family monarchy. Not a US I would want to live in, but it might be a fun place for a novel. What do you think?