It depends on the lens. For example 3N's kit lens SELP1650 min focus is 25cm

Yup that's about right. 25cm is still ok imo. Unless you're doing macro photography

Some compact will even go sub 1 cm.  but the sensor size is small and yes, image quality is not there at all.. 

You can go nearer if you are using old manual lens.. just shim with extension tubes.... cheap and nasty but works just a well unless you are the serious macro photographer.

The other issue is subject to distance magnification.. usually you will find a longer lens will work better for most situaiton than a shorter lens.. look around and you will see the 100mm range are about the best for macro work for APSC sensor because it give you 1:1 and you have good distance to camera.

Before I started changing my setup with full manual lens, I use my SEL55210 for macro work.. cant really get that close but a lot bigger image than I could with my kit lens.

Do note that when you get too close, exposure will become an issue... so are shadows. 

If you new and seriously into macro, do your homework before you consider compact.. there is quite a bit of workflow to understand. Each hurdle will have a new issue to overcome.  The solutions may not be expensive if you understand its limits. Most important is the base system you have selected.

simple extension will help on the macro shot!! but u must have manual control on the lens!!

Dslr sensor will produce better quality photos compared to compact. If you're new and really just want to take some pretty good quality photos at a budget, consider the panasonic lumix dmc-lx7. I think it can go as close as 1cm for macro shots. Check out the reviews.

But of course, once you step into dslr world, you may never want to go back to compact unless somehow portability and convenience is still more important to you

Some compact will even go sub 1 cm.  but the sensor size is small and yes, image quality is not there at all.. 

You can go nearer if you are using old manual lens.. just shim with extension tubes.... cheap and nasty but works just a well unless you are the serious macro photographer.

The other issue is subject to distance magnification.. usually you will find a longer lens will work better for most situaiton than a shorter lens.. look around and you will see the 100mm range are about the best for macro work for APSC sensor because it give you 1:1 and you have good distance to camera.

Before I started changing my setup with full manual lens, I use my SEL55210 for macro work.. cant really get that close but a lot bigger image than I could with my kit lens.

Do note that when you get too close, exposure will become an issue... so are shadows. 

If you new and seriously into macro, do your homework before you consider compact.. there is quite a bit of workflow to understand. Each hurdle will have a new issue to overcome.  The solutions may not be expensive if you understand its limits. Most important is the base system you have selected.

simple extension will help on the macro shot!! but u must have manual control on the lens!!

old manual lens price can be RM100 to RM1k++ and above depends on brand, model and conditions.
Extension tube is about RM30++ for a set of 3 tubes.
It can be up to very close depends on the lens.
this is without optical elements so it not affect the image quality much.

Don't buy those Magnifier type of Macro extension lens. it is quite useless with only can get sharp at the tiny center portion.

you can also use reverse macro ring which a lot of DSLR experts using.