Perhaps it is time to review some of the basics of Li-Ion cells...
Li-Ion cells do not naturally limit the charge they accept. Well, actually they do, through rapidly venting with flame, but we don't want to go there. They will continue to accept a charge until they blow.
The voltage they are charged to is directly related to their cycle life. 4.2 volts is accepted as a full charge, and it gives you around 500 cycles. If you charge to different voltage levels, you can influence your cycle life, and capacity. Charging to 4.1 volts gives you around 90% of your capacity, but your cycle life increases to above 1500 cycles.
Li-Ion cells oxidize from the inside. This oxidization begins upon manufacturer, so they have a limited calender life. The current chemistry being used is projected to have a calender life of 5 - 7 years. Earlier chemistry had a calender life of 2 - 3 years.
When you over discharge Li-Ion cells, the electrodes are dissolved by the electrolyte. Less electrode area results in less capacity. Foreign material that has been dissolved into the electrolyte can also create safety issues during subsequent charging.
When you over charge Li-Ion cells, the electrolyte breaks down. You can gain some extra capacity from the over charge, but cycle life is greatly reduced. For example, charging a single cell to 4.3 volts will give you around a 10% increase in capacity if you use the cell immediately. If you let it sit at that level of charge, you may not see this increase and the damage to the cell will increase. Charging to 4.3 volts will give you around 100 cycles. Charging to 4.4 volts is getting to the border where thermal run away begins. At that charge level cycle life drops to about 4 cycles if you use the cell immediately. If you store it at that level, it may rapidly vent with flame.
The reason you can't run down to the local store and buy Li-Ion cells is because they are dangerous. They are designed to be used in circuits that provide proper charging algorithms and also offer protection against over and under discharge. In addition, some batteries have an additional back up protection attached to the cells themselves. These protection circuits are designed to kick in if the main circuit fails for some reason.
The protected cells that are available have these types of protection circuits. They limit over charging to 4.35 volts and over discharging to around 2.5 - 2.8 volts. While neither of these limits is conductive to long cycle life, they do protect the cell from immediate destruction.
The critical key to using rechargeable batteries is to charge them with a quality charger. A quality charger follows the proper charging algorithm and charge rate for the chemistry being charged, and has proper termination.
Tom