The myths and stories get complicated because there are many different rechargeable chemistries (even "non-rechargeables" can be recharged under some circumstances - but you will need a specially built charger to do the job - ie. don't do it unless you fully understand what you are doing). Battery design has also tended to improve over time, although it is usually to increase energy densities and discharge currents.
The bottom line: Li-ion is one of the best (if not the best) rechargeable battery chemistries that are widely available. They do not have a memory in the same way as NiCd or even NiMH. But yes, they do "wear out" and will need replacing after a few years. In theory this should be less often with modern batteries compared to 10 year old Li designs.
You will find that an ageing Li-ion battery won't hold quite as much as when it was new, and it will self-discharge slightly quicker.
Don't worry about it.
Or perhaps if you can't help yourself, put a dollar in a jar every time you worry. You'll then have enough to replace the battery when the time comes (probably by buying an iPad 3)
Richard