Travelling to France with an expired German identity card

Certain agreements do exist (including with France), but there is no guarantee that travel is possible with an expired ID. Therefore the recommendation is to travel only with a valid travel document.

When leaving and entering Germany you are, as a German citizen, required to have a valid travel document (§1 PassG).

This can be a Passport, ID (Personalausweis) or a temporary identity card (vorläufigen Personalausweis).

Since you have no Passport and your ID has expired, you should apply for a temporary identity card.

Togeather with your expired ID, a biometric photo and for a fee (in Berlin) of €10 it should be issued immediately at your local registry office.

• Personalausweis vorläufig beantragen - Dienstleistungen - Service Berlin - Berlin.de
• Bundespolizei - Ersatzpapiere beantragen (in German)

Bei einem abgelaufenen Reisepass oder Personalausweis kann die Bundespolizei einen Reiseausweis als Passersatz für deutsche Staatsangehörige ausstellen. Dies gilt nur, wenn die Erteilung eines (vorläufigen) Reisedokumentes bei einer Passbehörde nicht mehr rechtzeitig zu erwarten ist.

In the case of an expired passport or identity card, the Federal Police can issue a travel document as a substitute for German nationals. This only applies if a (provisional) travel document is not expected to be issued in good time by a passport authority.

Personalausweisportal - Startseite - Corona Pandemie - Ausweispflicht, Personalausweis und Online-Ausweis
Ich muss verreisen. Welche Bestimmungen für Reisen ins Ausland muss ich beachten?
Deutschland hat mit einigen Europäischen Staaten vereinbart, dass deutsche Reisepässe oder Personalausweise bis zu einem Jahr nach Ablauf der Gültigkeit grundsätzlich als Identitätsnachweis anerkannt werden sollten. Zu diesen Ländern zählen unter anderem Belgien, Frankreich, Griechenland, Italien, Lichtenstein, Luxemburg, Malta, die Niederlande, Österreich, Schweiz, Slowenien.
...
Eine Reisegarantie ist mit diesem europäischen Abkommen jedoch nicht verbunden.
Um etwaige Schwierigkeiten bei der Reise mit abgelaufenen Dokumenten zu vermeiden, wird daher empfohlen, nur mit gültigen Dokumenten zu reisen.

I have to travel. What regulations do I have to observe when traveling abroad?
Germany has agreed with some European countries that German passports or identity cards should generally be recognized as proof of identity up to one year after their validity has expired. These countries include Belgium, France, Greece, Italy, Lichtenstein, Luxembourg, Malta, the Netherlands, Austria, Switzerland and Slovenia.
...
However, a travel guarantee is not associated with this European agreement.
In order to avoid any difficulties when traveling with expired documents, it is therefore recommended to only travel with valid documents.

§1 Passport requirement - (PassG)
(1) Germans within the meaning of Article 116 (1) of the Basic Law of the Federal Republic of Germany leaving or entering the geographical area in which this law applies are required to carry a valid passport to identify themselves. Presenting a passport of the Federal Republic of Germany within the meaning of (2) shall fulfil this requirement.

§3 - Temporary identity cards - Act on Identity Cards and Electronic Identification (Personalausweisgesetz, PAuswG)
(1) A temporary identity card shall be issued to any applicant who provides a credible reason for immediately requiring an identity card.

One important thing is that in France, unlike Germany, there is no obligation to carry or even hold any specific type of identification document. And as an EU citizen, you also have very extensive rights to visit France. So you are not committing an offense and do not risk a fine.

What could happen:

1. Entering with a recently expired travel document is common and accepted. You will find mentions of this in airline databases (at least for passports) and I have personally made use of it at the Swiss border, back when it was actively policed. As you know, you're also extremely unlikely to be challenged at the border. Border guards are also aware of your rights as an EU citizen and you're not a priority for them.

2. In the unlikely event you are asked to provide ID, the police will complain a bit, maybe try to lecture you about the requirement to carry a non-expired ID (they tell that even to French citizens where it is explicitly and unambiguously untrue) but since you have not committed an offense and they know it, they will let you go after a few minutes. Still be apologetic about it, there is no point in confronting a police officer.

   The most the police can conceivably do is keep you for a few hours to ascertain your identity and citizenship but it doesn't make sense for them to do that. If the police is checking ID, it's either because they really want to know your name (and then your ID card is enough for that) or because they want to pressure people they actually know or find low hanging fruits to boost stats on forced removals and the like. If you have a recent ID, they know you're an EU citizen so this won't result in any of that and dealing with you any further would just be a hassle.

3. The hotel may indeed ask to see your national ID card. Legally I don't think the treaty is terribly relevant here but if needed, you can tell them about what you read on the ministry of foreign affairs website. They just need an excuse to feel good about it so offer it to them. Just don't be obnoxious about it (more like “oh, I thought blablabla, I am very sorry” than ”this is my God-given right, call your manager immediately”). They also most likely won't check or notice the date, they are scanning the document for your name.

The return to Germany is another matter that has been addressed in another answer and what I would be most concerned about. You cannot be refused entry but both the letter of the rules and the general approach to their enforcement do make a fine a real possibility.