Good AA rechargeables ones now give similar capacity to alkaline, around 2.8Ah. At £1.50 a cell lasting 500 charges that's 0.3p per charge. Alkaline AAs at 17p each are thus 56x the cost.
Zinc carbons are much cheaper per battery, but give even less capacity per cost.
The cost of the recharging electricity is trivial. A 2.5Ah 1.2v cell holds 3 watthours, or 0.003 kWh. Allowing for inefficiencies, 0.005kWh costs 0.005x13p = 0.065p. You can recharge 15 for a penny.
- Alkaline cost: 17p each
- Rechargeable cost plus electricity: 0.365p
Where capacity isn't so important, cheap rechargeables at half the price typically give around 1/3 the capacity.
When they're not as good
With LED devices such as torches or remote controls, rechargeables' lower voltage reduces light output or range, making the batteries run longer.
Rechargeables don't last as long in very low drain apps such as clocks, due to gradual self discharge.
There are still some appliances that won't work with rechargeables, such as a minority of cameras.