Are there any insurances in Germany that are considered important in Germany but which are not required to have as for example a health insurance?
In Germany, the following insurances are recommended and a big fraction of the population has such insurance:
Haftpflichtversicherung: This insurance will cover damages that you cause to others such as ruining someone's couch by pouring wine over it. It will normally also cover for damages of your kids as long as you or the damaged person can prove that you did not properly watch over them while they were causing the damage. (Otherwise, your kids would be responsible for the damage themselves but would not have to pay for the damage because of their young age, i.e. the damaged person will not be refunded.)
Berufsunfähigkeitsversicherung: This insurance pays you a given amount when you get injured such that you are no longer able to work for a longer period / life. This insurance is recommended since the state coverage for people that cannot was reduced to a minimum amount some years ago.
Auslandskrankenversicherung: A public health insurance will only cover the costs of health care to the amount they would cost in Germany. This means that you will have to pay a significant amount yourself when receiving treatment on travels in countries with expensive health care like the United States. This insurance will pay the difference or even the entire amount, as long as you did not travel to another country with the purpose of receiving treatment. Some insurances also exclude a person's home country if one was not born in Germany to avoid fraud and overly expensive customers that are likely to be treated abroad a lot.
Other than that some people have additional insurance for their teeth when insured with a public health insurance. Pflegetagegeldversicherungen which pay a given amount if you need treatment in a care home are also gaining popularity. Finally, it is worth mentioning Unfallversicherungen which pay you a certain amount on accidents that cause damage to your body and will normally also cover, for example, rescue from a mountain or the sea.