Nowadays, one practical limitation of deep neural network (DNN) is its high degree of specialization to a single task or domain (e.g., one visual domain). It motivates researchers to develop algorithms that can adapt DNN model to multiple domains sequentially, meanwhile still performing well on the past domains, which is known as multi-domain learning. Almost all conventional methods only focus on improving accuracy with minimal parameter update, while ignoring high computing and memory cost during training, which makes it impossible to deploy multi-domain learning into more and more widely used resource-limited edge devices, like mobile phone, IoT, embedded system, etc. During our study in multi-domain training, we observe that large memory used for activation storage is the bottleneck that largely limits the training time and cost on edge devices. To reduce training memory usage, while keeping the domain adaption accuracy performance, in this work, we propose Deep Additive Attention Adaption, a novel memory-efficient on-device multi-domain learning method, aiming to achieve domain adaption on memory-limited edge devices. To reduce the training memory consumption during on-device training, $DA^3$ freezes the weights of the pre-trained backbone model (i.e., no need to store activation features during backward propagation). Furthermore, to improve the adaption accuracy performance, we propose to improve the model capacity by learning a novel additive attention adaptor module, which is also designed to avoid activation memory buffering for improving memory efficiency. We validate $DA^3$ on multiple datasets against state-of-the-art methods, which shows good improvement in both accuracy and training time.