Additionally the materials which make up the battery can vary. The cathode of a Li-Ion battery is typically some type of lithium intermetallic compound (typically Lithium-Cobolt based) and the anode can vary (from simple graphite to intermetallic compounds to more exotic things like Sn nano-particles, polymers etc).
In 2010 my undergraduate thesis in materials engineering was on developing a prototype anode made from graphene dispersed Tin-Cobolt nanoparticles. This was back when graphene was first discovered and was being pretty widely hyped as a 'wonder material'. It was hoped that by dispersing the metal particles in graphene matrix we could better accommodate the mechanical stresses that come from cycling the battery (when you intercalate and deintercalate Li ions there is a volume expansion and subsequent contraction - this is what causes battery to degrade after charge/discharge cycles). Results were mostly inconclusive largely due to unreliability of synthesis technique we used for manufacturing graphene - Electrostatic Spray reductive Precipitation (ESRP).
But yes Undergraduates are messing around with Li-Ion technology its not surprising corporations are too.