Customer lifetime value (LTV) prediction is essential for mobile game publishers trying to optimize the advertising investment for each user acquisition based on the estimated worth. In mobile games, deploying microtransactions is a simple yet effective monetization strategy, which attracts a tiny group of game whales who splurge on in-game purchases. The presence of such game whales may impede the practicality of existing LTV prediction models, since game whales' purchase behaviours always exhibit varied distribution from general users. Consequently, identifying game whales can open up new opportunities to improve the accuracy of LTV prediction models. However, little attention has been paid to applying game whale detection in LTV prediction, and existing works are mainly specialized for the long-term LTV prediction with the assumption that the high-quality user features are available, which is not applicable in the UA stage. In this paper, we propose ExpLTV, a novel multi-task framework to perform LTV prediction and game whale detection in a unified way. In ExpLTV, we first innovatively design a deep neural network-based game whale detector that can not only infer the intrinsic order in accordance with monetary value, but also precisely identify high spenders (i.e., game whales) and low spenders. Then, by treating the game whale detector as a gating network to decide the different mixture patterns of LTV experts assembling, we can thoroughly leverage the shared information and scenario-specific information (i.e., game whales modelling and low spenders modelling). Finally, instead of separately designing a purchase rate estimator for two tasks, we design a shared estimator that can preserve the inner task relationships. The superiority of ExpLTV is further validated via extensive experiments on three industrial datasets.