What is the biggest tire I can run on my factory Chevy Tracker 15 inch alloy rims?  They are only 5.5 inches wide.

They will shoe horn on, I've done it, but the shop might not mount them for you.

I had a 33 12.5 on a stock rim  , changed it out to a 7" RockCrawler

What brand of tire and size looks the best and works the best on the factory five and a half inch wide wheels?

You should be able to run 235 75 R 15's or 30 9.5 15 on them on the big side of the spectrum...

anopther option is to search the pick-n-pulls/Thrifty Nickel for full-sized Bronco "wagon wheels".  They look very much like the Samurai "Wagon Wheels" except they are much wider. 

I have a set with 30x9.0 BFG AT/KOs on my Samurai.  I got a set of 4 for 40 bucks...

I've ran 235/75&70/15 series tires on my trucks with various stock 5.5" rims from 1st gen models to second gen models, aluminum and steelies.  I've experienced no problems to date.   I'm talking collectively easily over a half-million miles to date since 1994.

I had a set of 235/70/15s LTX Michelins mounted on one of my Trackers 2 years ago.  I was driving an interstate in February, hit a patch of ice  and began to slide sideways down the interstate at about 55MPH.  I countered the slide but I slid sideways down the interstate about a hundred feet or so and then into the sloped grassy meridian, sideways.  Fortunately I did not flip and the tires remained on the wheels with no damage EXCEPT that dirt was forced into the rim/tire seal lip and caused the two tires on the slide side to leak air until I had them removed, cleaned and remounted.  The error wasn't the tire... it was the operator (me).

The 30X9.50X15 profile isn't much different from the 235 and the actual height rarely is a true 30" diameter... so I personally don't consider them any more riskier than the 235 series tires on a 5.5" rim.  Concurrently, for me, since they aren't a full 30" tire (usually 29.2 or 4" diameter) I don't consider the extra cost worth it over a 29" - 235 series tire, either.  Then there is the other school of thought that humans can, do and will break an anvil with a rubber mallet so I guess it is proportional to the operator.

Ack-
I did exactly what you said and found on CL a pair of Goodyear GSA 30x9.50x15 on old Bronco white steel wheels for $40, they were dirty but 2 had half tread and 2 had a third tread.  I may use them around town or on highway runs over my fat boys, anyway you look at, now I have rollers and wheels I can sacrifice at paint time, plus they will get used on another project I am dreaming of.  I may just bolt them on or I may try mounting the rubber on my factory alloys if I get a decent quote on mounting them. Thanks for the input.

i run 31x10.5's on stock sidekick wheels, looking for some of the tracker ones that are 7"wide though...

but they do fit on the rims, give a good amount of side lug and just look plain mean.

Hello All,

I just mounted 31x10.5R15 on the stock rims.

I had 235/75/15s on my stock sidekick wheels, it worked ok but they were too narrow!
My 235/75/15 are 28.7" tall and the 30/9.50/15 I was looking at were 29.7" tall. I have the 235 tires on 15x7 rims and they
Look much better and are wearing better too. Anything can be done!

Drone637 is running 30X9.50X156's on his Trouble Race Car.

I just got back from mounting my new Toyo Open country 225/75 15s and they look great.  They are 28.3" and are probably a safe bet if this is a daily driver too.

The tire manufacturer will specify a range of rim widths for any given tire, and it does vary from one tire to the next based on the construction - so pick a tire and look it up...  

^^ what he said!

Just 'cause it can be made to fit in the shop, does NOT mean it's a good idea out on the highway or miles into the backcountry.  I do not presume to know more than the tire manufacturer's engineers, so I choose to follow what they recommend, for a reason. How much do you really want to trust to what some unknown poster once claimed on the internet:  "Well, it worked for me!"  
The range of rim widths that BFG specifies for my all-terrain TA/KO tire in the 30x9.5 size is 6.5 - 8.5".  The largest tire size BFG specifies for a 5.5" rim is 215/75R15.  Again, go look it up!  

my wife runs 31 10.50 on her 99 vintara.with 1.6 and gets 21mph. with a 5 speed.