Would I be able to charge my car battery with a DC power supply that outputs 12V 2.5A? I know real car battery chargers put out 12V 10A. Would it just take 5 times longer?
I know real car battery chargers put out 12V 10A.
There's your problem. A 12 volt lead-acid battery charger must put out (about) 14 volts. Google lead acid battery charging.
A LA battery will not charge unless supplied with a voltage 1V above battery voltage, or more. You can charge your battery to 11V (1.8V per cell). At 11V it is 'empty', so 12V is inadequate. A good charge voltage is 13.8 (2.3V per cell), 14.4V if you are in a hurry.
You can attach a boost converter to that 12V power supply and use 13.8V to charge your battery, occasionally disconnecting it, waiting a minute, measuring battery voltage with a multimeter. When the battery is at 12.8V or more, it's 'full' - then stop. Google 'float charge 12v battery'. It's fairly safe to float charge at 13.6V for longer periods of time. So yes, for 5$ extra you can charge that battery at 25W (assuming an efficiency of 85% on the boost converter).
Disregard the rest of this answer if you don't love LA batteries like I do.
There is an argument to be made for using variable voltage to charge. I have used a Black & Decker charger which did this very nicely, on an old battery, starting at 10V, and worming up to 14V. The same device can be made with an arduino (or similar, needs an ADC), a 10V zener diode, a MOSFET, and a max 14.4V voltage input. At high currents, the Vgs_on of a MOSFET must be correct (defined as Vgss in the datasheet), or it will overheat (being a resistor at heart), even explode. That's the only rotten easter egg in this process. It will cost a pittance.
My first version of this didn't have variable voltage with PWM, but it should have had this. Using PWM, you can simply control the voltage that reaches the battery terminals, from 0% to 100%. You can use an NPN+PNP pair on the zener's path to prevent leakage when not measuring battery voltage. It is not strictly necessary to use the C/1, C/5, etc. charging schemes, but you should probably install a fuse somewhere.