For solar trickle charging stick with lead acid batteries for a trouble-free system, or be prepared to spend a lot replacing batteries.
NiCds don't like trickle charging (with DC) as it encourages dendrite formation;
Lithium Ion batteries have a high energy density but short lifespan (and they are dangerous if not charge-managed properly). Unless you know what you are doing, stay well away - they can explode or catch fire if charged or used inappropriately,
Car batteries are designed to deliver short bursts of high current (starting) with immediate recharge - they don't like applications with a long continuous energy drain between recharges, even if that drain is small. Lead-acid "leisure" batteries are built differently. They cope better with deep discharge too (it damages car batteries). Both sorts eventually chemically 'wear out', but leisure batteries last far longer, and a good lead acid design will sit for years at full charge without harm (as seen in emergency lighting, alarm systems, telephone exchanged and standby generator sets).
As a general rule, lead acid should be kept very close to full charge. If it's a battery you will rely on, make sure it can be serviced. By that I mean that you can refill with distilled water as necessary. You don't want a sealed one.
If you are using unregulated ('dumb') charging, you also need to keep the battery in a well ventilated place. When fully charged, further charge voltage causes 'gassing' -- you electrolyse the water in the battery, breaking it down into hydrogen and oxygen. It's an explosive mixture. Commonly the hydrogen simply escapes (it's a lot lighter than air and H2 is the smallest molecule so it passes though many materials), but an oxygen-rich atmosphere is also dangerous.
Electrolysing the water is the reason why you need to top-up with distilled water occasionally, to replace what's been lost. Intelligent chargers can minimise this, but it's probably impossible to eliminate it.
Hope that helps.
E.
