Ionospheric structure usually changes dramatically during a strong geomagnetic storm period, which will significantly affect the short-wave communication and satellite navigation systems. It is critically important to make accurate ionospheric predictions under the extreme space weather conditions. However, ionospheric prediction is always a challenge, and pure physical methods often fail to get a satisfactory result since the ionospheric behavior varies greatly with different geomagnetic storms. In this paper, in order to find an effective prediction method, one traditional mathematical method (autoregressive integrated moving average—ARIMA) and two deep learning algorithms (long short-term memory—LSTM and sequence-to-sequence—Seq2Seq) are investigated for the short-term predictions of ionospheric TEC (Total Electron Content) under different geomagnetic storm conditions based on the MIT (Massachusetts Institute of Technology) madrigal observation from 2001 to 2016. Under the extreme condition, the performance limitation of these methods can be found. When the storm is stronger, the effective prediction horizon of the methods will be shorter. The statistical analysis shows that the LSTM can achieve the best prediction accuracy and is robust for the accurate trend prediction of the strong geomagnetic storms. In contrast, ARIMA and Seq2Seq have relatively poor performance for the prediction of the strong geomagnetic storms. This study brings new insights to the deep learning applications in the space weather forecast.