Are you thinking of any specific time periods/locations?

I have a book about early Danish kings and their families it gives this list:

• Svend Estridsen (Sweyn II in english)- 19 illegitimate children
• Harald Hen (III) - 0 children
• Knud den Hellige (Canute the Saint) - no illegitimate children
• Oluf Hunger (Olaf Hunger) - no illegitimate
• Erik Ejegod (Eric Evergood) - 4 illegitimate children
• Niels - 1 illegitimate daughter
• Erik Emune (Eric the Memorable) - 1 illegitimate son
• Erik Lam (Eric lamb) - 1 illegitimate son
• Svend Grathe (Sweyn Grathe) - no illegitimate children
• Knud magnussen (Canute Magnussen) - 6 ilegitimate children
• Valdemar den Store (Valdemar the Great) - 1 illegitimate son

Note that these are the acknowledged bastards, I.e children from "Friller" or concubines. Children that were the results of secret affairs, do not generally show up as such in historical documents, whether or not it was known at the time.

My main source is a Dansih book, I've found This site in English it's the same list translated to English.

Quick Edit: This is a much better source, with sources for each king. My list begins at 3. B

Further Edit: After writing that I'm realising it doesn't really answer the question. I do think it goes to show that there isn't really a "normal number, but that very high numbers certainly did occur.

Ten is actually a bit of a lowball depending on where you're talking about, the time period and what "level" of nobility they were at! The English kings through history have been fairly good at times about tracking their illegitimate offspring:

Henry I had somewhere around 25 bastards, give or take five.^^[1]

Charles II had 14 acknowledged illegitimate children but the numbers overall are believed to be twenty or more.^^[2]

William IV, who ruled over the United Kingdom, had 11 that we know about.^^[3] When Queen Victoria took the throne, the open acknowledgement of royal bastards essentially stopped because she banned them from court.^^[4]

Eclipsing everyone, however, at least by reputation, is August II of Poland. It is not actually known exactly how many illegitimate children he had and the estimates versus the ones he acknowledged are incredibly disparate, with estimates that I've seen going up into the range of the mid-three hundreds. The general consensus seems to be "probably a lot," which is not really exact but it is worth noting that he was noted for the number of illegitimate children he had at a time when that sort of thing was not particularly unusual.^^[5]

~

^^[1] ^^Kathleen ^^Thompson, ^^"Affairs ^^of ^^State: ^^The ^^Illegitimate ^^Children ^^of ^^Henry ^^I". ^^Journal ^^of ^^Medieval ^^History ^^Volume ^^29, ^^Issue ^^2, ^^June ^^2003, ^^pp. ^^129–151.

^^[2] ^^George ^^Edward ^^Cokayne, ^^The ^^complete ^^peerage ^^of ^^England, ^^Scotland, ^^Ireland, ^^Great ^^Britain, ^^Vol. ^^6

^^[3] ^^[4] ^^Peter ^^Beauclerk ^^Dewar ^^& ^^Roger ^^Powell, ^^Royal ^^Bastards.

^^[5] ^^Sara ^^Eigen ^^Figal, ^^Heredity, ^^Race, ^^and ^^the ^^Birth ^^of ^^the ^^Modern

Follow up question, are there known records of a European queen giving birth to a bastard?

The 15th century French King Charles VII had a mistress named Agnes Sorel, who gave birth to three of his daughters and was pregnant with a 4th before dying under mysterious circumstances, perhaps poisoning. She is famously depicted in one half of a diptych wing by the French artist Jean Fouquet, known as the Melun Diptych. Yes, she is depicted as the Virgin Mary enthroned in Heaven and yes, she has unusual-looking breasts (typical with "Virgo Lactans" iconography, so that people won't go around thinking naughty thoughts about the Virgin Mary nursing).