How many times can I upgrade my lightning spell?

The maximum level you can obtain in Research Lab Level 3 is level 4(for your lightning spell). If you want your Lightning spell to be leveled up to level 5, you must upgrade your Research Lab to level 4.

As BiscuitBaker stated, the highest level you can get on Lightning spell is level 6(at least for right now).

Source

At Lab level 3, the maximum level you can research for Lightning is 4.

Overall, the maximum level for the Lightning spell (currently, at least) is 6:

http://clashofclans.wikia.com/wiki/Lightning_Spell

Though you'll need the eighth level lab (requires Town Hall 10) to get that far.