Please, keep in mind that as the experience is scaled based on your level, this method will be far less effective with high level characters. (One of the fastest ways from fifty-nine to sixty is probably the experience boost from becoming a spectre at the beginning of the game).
The following approaches exploit a glitch in the save system which allows the player to receive experience after loading a previously saved game file. There are several areas where saving and reloading [that same] game file resets the particular trigger intended to reward experience points. Deleting the autosave file places your most recent save at the top of the "Load" list, thus making it the default selection. Saving the extra keystroke will allow you to complete an entire save and load cycle in under thirty seconds. Saving to a new file each time will save you an additional motion and allow you a plethora of backups but mind the amount of free space on your harddrive as the files can be large
*Found in the Prothean ruins located in Artemis Tau / Knossos / Therum. After clearing the cavern of Geth the player activates the mining laser (combination: A, X, B, Y) and travels up an elevator to the floor on which Liara is imprisoned. The player will receive experience points [scaled to increase relative to your current level] after stepping off the elevator platform and onto the blue tile. Save immediately into a new slot (in case you need a prior load for backup purposes)-- it is critical to save before speaking with Liara. After the save is complete, proceed to the "Load" menu and select the file you just created. You will receive experience again upon load completion. Save again, then load. Rinse and repeat.
*A terminal inside the Feros freighter can be hacked after each save, load cycle.
*The lock on the sewer door in the Virmire base can be picked after each save, load cycle.
*The assault on the Citadel forces the player to exit the elevator. There are several hallways in following section where this method can be used to exploit the high respawn rate of Geth Destroyers.
//Please, feel free to add comments or make suggestions so I may update this!
[my writing; also posted to Mass Effect Wiki]