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New malware emerges at a rapid pace and often incorporates Domain Generation Algorithms (DGAs) to avoid blocking the malware's connection to the command and control (C2) server. Current state-of-the-art classifiers are able to separate benign from malicious domains (binary classification) and attribute them with high probability to the DGAs that generated them (multiclass classification). While binary classifiers can label domains of yet unknown DGAs as malicious, multiclass classifiers can only assign domains to DGAs that are known at the time of training, limiting the ability to uncover new malware families. In this work, we perform a comprehensive study on the detection of new DGAs, which includes an evaluation of 59,690 classifiers. We examine four different approaches in 15 different configurations and propose a simple yet effective approach based on the combination of a softmax classifier and regular expressions (regexes) to detect multiple unknown DGAs with high probability. At the same time, our approach retains state-of-the-art classification performance for known DGAs. Our evaluation is based on a leave-one-group-out cross-validation with a total of 94 DGA families. By using the maximum number of known DGAs, our evaluation scenario is particularly difficult and close to the real world. All of the approaches examined are privacy-preserving, since they operate without context and exclusively on a single domain to be classified. We round up our study with a thorough discussion of class-incremental learning strategies that can adapt an existing classifier to newly discovered classes.