What are the top must have accessories for a new biker?

2 x spare tubes, 2 x tyre (tire) levers, tyre boot, Pump. Bottles & Cages for the bike.
Helmet, Eye protection & track mitts, Shorts/tights & shoes for you.
Enjoy your bike and ride lots and safely.

A good set of clipless pedals is always a great thing to have.

My priorities, more or less in order, would be:

helmet
shorts (two pair minimum: one to wear, one in the wash)
gloves
bottles and cages
seat pack with spare tube, patch kit, inflator, maybe a multi-tool
cyclocomputer
stiff-soled shoes and clipless pedals
jerseys
bike-specific tools, if you want to do your own wrenching
rain gear and/or cold-weather clothing

IMO you need the first three items for even the shortest of rides. Add the bottles, cages, and seat pack by the time you work up to rides of an hour or more, and/or venture more than a few miles from home. From there on it's largely a matter of what kind of riding you do and personal preferences. Most people tend to build up their collection of jerseys, tools, and rain/cold gear over the years.

What are the top must have accessories for a new biker?

#2-Stuff to fix flats.

#3- Fenders.

#4- A way to carry things including water.

#5- Lights and reflectors (the reflectors that come with your bike are very low quality).

#6- A bell.

#7- Cycling clothing for the parts of you that touch the bike.

#8- A helmet.

#9- Sunscreen.

#10- Clipless pedals if you don't already have them.

A decent floor pump.

Len Zinn's book: Zinn & the Art of Road Bike Maintainence

I wouldn't consider a helmet an accessory--it's a necessity. I've fallen many times in 30 years, and I think I've only hit my head once, but I still don't go down the driveway without one. Ditto a water bottle, at least for rides of more than about 20 minutes, and the tool kit with patches and pump (are there really people who ride WITHOUT that?) I average a flat about every 100 miles here in Big Thorn Country--I'd be walking every second or third day. Here in the desert, anyway, I'd put sunscreen on the necessity list as well. I've been riding without it all my life, and the last few years I've had to have several "probably sun-related lesions" removed from my arms and legs. Had a nice tan back there in the '80s, though.
Now that I think of it, a lot of the rest of the stuff mentioned here, like shorts and shoes and gloves, are pretty much necessities, too. First thing I'd put on the "nice, but not necessary" list is probably a computer. You don't need it, and I've ridden thousands of miles without one, but they're nice to have.

No one mentioned this, but I consider a cell phone a necessity. We had someone on a group ride have a (fortunately) minor heart attack and were glad we could call for help. Also very useful for less dramatic events (my wife loved when I called for a pickup when my chain failed). In the end, very few problems that can't be helped by a timely call.

And most phones come with an 'off' button if you don't want to receive calls.

One other item that is useful where I live is a bicycle train pass. If caught in a bad storm I can usually get to a train line that can get me close to home.

I agree with alll the above, but would include a mirror(either helmet or bike mounted). It has saved me many times in an around traffic.

Brian