You can start a so called €1 limited liability company in less than 10 days all over the EU incl. Germany.
- I assume you're referring to founding a UG with 1€ in Germany. If you truly found it with 1€, it'll technically be bankrupt shortly after founding it because the founding costs are around 1k€ (notary etc.), and you haven't made any revenue yet. It's generally recommended to found with at least 2-3k€.
- 10 days is not a reasonable timeframe for founding a German UG. Optimistic timeline: Instant notary appointment, 14 days for corporate registry, instant bank account, 14 days for tax ID. Total of 28 days.