Deep learning has achieved great success in recognizing video actions, but the collection and annotation of training data are still quite laborious, which mainly lies in two aspects: (1) the amount of required annotated data is large; (2) temporally annotating the location of each action is time-consuming. Works such as few-shot learning or untrimmed video recognition have been proposed to handle either one aspect or the other. However, very few existing works can handle both issues simultaneously. In this paper, we target a new problem, Annotation-Efficient Video Recognition, to reduce the requirement of annotations for both large amount of samples and the action location. Such problem is challenging due to two aspects: (1) the untrimmed videos only have weak supervision; (2) video segments not relevant to current actions of interests (background, BG) could contain actions of interests (foreground, FG) in novel classes, which is a widely existing phenomenon but has rarely been studied in few-shot untrimmed video recognition. To achieve this goal, by analyzing the property of BG, we categorize BG into informative BG (IBG) and non-informative BG (NBG), and we propose (1) an open-set detection based method to find the NBG and FG, (2) a contrastive learning method to learn IBG and distinguish NBG in a self-supervised way, and (3) a self-weighting mechanism for the better distinguishing of IBG and FG. Extensive experiments on ActivityNet v1.2 and ActivityNet v1.3 verify the rationale and effectiveness of the proposed methods.