It depends on what you call an exception. Both Queen Mary II and her sister Queen Anne were born granddaughters in the male line of a King and the daughters of the reigning monarch's heir but they were often referred to as Lady and not always styled as princesses until they married princes.
So Royal titles were not that extensive in the past and gradually got extended especially in the 19th century.
Two factors play a role in restricting titles and styles these days:
1 more distant Royal relatives want to live a more private and ordinary life. They don't get funded by the state and need to earn their keep being easily recognisable as a Royal means they get unwanted attention and possibly rumours or doubts on how they got the job they have to earn their keep. The more limited the amount of Royals is the less chance of scandal caused by a distant relative is another part of this desire for restricting titles.Societies have become less hierarchical and special benefits based on the fact that you descent of a monarch through some junior line a few generations ago simply no longer is acceptable. It was in previous centuries but not these days.
The second factor is that where in the past lines were often limited or even ended by higher mortality rates, male or male preferred succession and excluding heirs for marrying beneath their status. That has changed with Royals marrying for love and gender neutral succession as well as improved health care. Take Belgium if all four of king Philippe's children as well as all 8 of his nieces and nephews would all have children styled as HRH prince(ss) of Belgium the Royal family would quickly grow within a few generations from a relative small number to dozens or more.
Still it's not all that new. At the end of the 18th century the Stadholder prince Willem V of Orange had three children. A daughter Louise and two sons his heir Willem Frederik, hereditary prince of Orange and prince Frederik.
After the heir married his Prussian first cousin by birth an HRH the stadholder suggested it would be better if his younger son would not marry a Royal princess but an ebenburtig lady from a lower rank than a Royal house. So the two sisters-in-law would not get in to a potential conflict about status and style.
The younger prince did fall in love with a Royal princess (princess Mary daughter of George III) but as he was exiled at that time he was in no position to propose to her. He went into service of the Austrian army and died shortly afterwards in Italy. But it already shows that even in the 18th century rulers were thinking about the status of younger lines and how rivalry with the senior line might not be desirable.
Has restricting titles and styles [e.g. the Letters Patent of 1917, issued by King George V of Great Britain, who restricted the title Prince(ss) of the UK with the qualification of Royal Highness to children and grandchildren through sons of British monarchs; Czar Paul, who restricted the title Grand Duke/Duchess of Russia with the qualification of Imperial Highness to children and grandchildren through sons of sovereigns, the title Prince(ss) with the qualification to male-line great-grandchildren of sovereigns, and the title Prince(ss) with the qualification of Serene Highness to male-line great-great-grandchildren of sovereigns] actually been the exception, not the rule, throughout European royal history?
Until, that is, recent times, when the kingdoms of Norway and Netherlands have also streamlined membership in their royal houses and restricted titles? And Great Britain has been further limiting titles (look at the children of the Earl of Wessex), although I believe all the children of the Duke of Cambridge are styled as Prince(ss), with the qualification of Royal Highness. Normally, Charlotte and Louis would be only Highnesses, like the children of Prince Joachim of Denmark ...
Sweden and Belgium at first expanded titles; but now it looks like they're going in the opposite direction.
Otherwise, it seems that titles and styles have generally been unlimited in dynastic male lines: this certainly has been the case in all the German-speaking houses. Liechtenstein, for instance, is teeming with princes and princesses: the country has far many more of those than it does square miles.
Even in the Scandinavian countries, it seems that titles and styles have likewise been unlimited in dynastic male lines. If in the past one had a great-grandchild of a monarch styled as only a Count(ess), it typically was because of a morganatic marriage. Otherwise, one has a situation like the Greek royals, who are princes and princesses of Denmark as male-line descendants of King Christian IX. They are Royal Highnesses, of course, as princes and princesses of GREECE only -- since Danish royals descended from younger sons of sovereigns normally don't get styled as HRH.
Elsewhere, outside the German speaking houses, one has that non-reigning members of even a mere DUCHY like Parma were styled as Prince(ss), with the qualification of Royal Highness, as male-line descendants of King Louis XIV of France. And in the grand duchy of Luxembourg today, male-line descendants of the Grand Duchess Charlotte and her consort (Prince Felix of Bourbon-Parma) have likewise been Royal Highnesses. Before, non-reigning members were only Grand Ducal Highnesses.
And I believe titles and styles have also been unlimited in dynastic male lines of the various French houses -- e.g. royal Orléans and imperial Bonaparte, not to mention other offshoots of the Bourbons (such as the aforementioned Parma) and the kingdom of Two Sicilies.
But back to my original question: has restricting titles and styles, then, generally been the exception throughout European royal history (given what one has observed in other houses, both German and non-German)?