but can't the alternator fully re-charge the battery as well if you drive for a long period of time?

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Current cars:
2001 MB E55 AMG
2003 Honda S2000
2000 Honda CRV

Gone but not forgotten:
2004 BMW 325i 5sp sport
1999 BMW 540i sport
1997 MB E420
1995 BMW M3 coupe