To operate, LED needs more than 1.5V voltage level. If you want to operate a LED with a single cell battery, you must boost the voltage to enable powering the LED. Here you can use two transistor circuit for boosting the battery’s voltage:

Two Transistors Single Cell Battery LED Driver circuit schematic

The transistor shown in the schematic diagram is 2N3904 and 2N3906, but you can use other transistor as well, as long as the current handling capability and the amplification is similar (or higher). You can explore the circuit by changing L1 and C1 values to optimize the circuit for best efficiency.