First of all, are there magazine limits in Illinois? 

In Illinois state law there are NO magazine limits or capacity restrictions on pistol or rifle magazines. There are however local ordinances (read more below) that may regulate magazine capacity limits, but these ordinances apply only to rifle magazines because the FOID Act & the Firearms Concealed Carry Act both preempt local regulations of handguns for FOID and/or CCL holders.

FOID Act

Sec. 13.1. Preemption.

    (a) Except as otherwise provided in the Firearm Concealed Carry Act and subsections (b) and (c) of this Section, the provisions of any ordinance enacted by any municipality which requires registration or imposes greater restrictions or limitations on the acquisition, possession and transfer of firearms than are imposed by this Act, are not invalidated or affected by this Act.

    (b) Notwithstanding subsection (a) of this Section, the regulation, licensing, possession, and registration of handguns and ammunition for a handgun, and the transportation of any firearm and ammunition by a holder of a valid Firearm Owner's Identification Card issued by the Illinois State Police under this Act are exclusive powers and functions of this State. Any ordinance or regulation, or portion of that ordinance or regulation, enacted on or before the effective date of this amendatory Act of the 98th General Assembly that purports to impose regulations or restrictions on a holder of a valid Firearm Owner's Identification Card issued by the Illinois State Police under this Act in a manner that is inconsistent with this Act, on the effective date of this amendatory Act of the 98th General Assembly, shall be invalid in its application to a holder of a valid Firearm Owner's Identification Card issued by the Illinois State Police under this Act.

Basically what this means is that when the Illinois Concealed Carry Act was passed it was written into it's language, and amended into the FOID Act, that ONLY the state can regulate handgun laws. Therefore local regulations for pistols are null and void. Bottom line: If you possess a pistol with a valid FOID or CCL card, you are doing so under those acts and State authority, not local, and as of this writing, there are no State magazine limits for handguns.

What are the magazine limits in Cook County? 

For a FOID or CCL holder, there are NO valid magazine limits or capacity restrictions for handgun magazines in Cook County Illinois due to preemption by State law. The magazine limit for rifles is 10 rounds though due to the Blair Holt Assault Weapons Ban. 

Sec. 54-212. - Assault weapons, and large-capacity magazines; sale prohibited.

    (a) It shall be unlawful for any person to manufacture, sell, offer or display for sale, give, lend, transfer ownership of, acquire, carry or possess any assault weapon or large capacity magazine in Cook County

   (definition) "Large-capacity magazine" means any ammunition feeding device with the capacity to accept more than ten rounds, but shall not be construed to include the following:

(1) A feeding device that has been permanently altered so that it cannot accommodate more than ten rounds.
(2) A 22-caliber tube ammunition feeding device.
(3) A tubular magazine that is contained in a lever-action firearm.

Can you have 15 round magazines in Cook County then?

Yes. As long as you have a valid FOID and/or CCL and the magazines are for a pistol.
'High capacity' rifle magazines are not legal, but as far as we have been able to discern no FOID or CCL holder has been convicted of violating Sec. 54-212 in regards to large capacity magazines. However, not all
such convictions would be public record.
The language of 'large or high capacity' has been challenged in court though. 

What are the magazine limits in Chicago? 

For valid FOID or CCL holders, there are NO magazine limits or capacity restrictions for handgun magazines in Chicago Illinois due to preemption, same as within Cook County overall. The magazine limit for rifles is 15 rounds though due to the City of Chicago Municipal Codes.

8-20-085 High capacity magazines and certain tubular magazine extensions – Sale and possession prohibited 

   (a)   It is unlawful for any person to carry, possess, sell, offer or display for sale, or otherwise transfer any high capacity magazine or tubular magazine extension for a shotgun. 

   (definition)  "High capacity magazine” means a magazine, belt, drum, feed strip, or similar device, including any such device joined or coupled with another in any manner, that has an overall capacity of more than 15 rounds of ammunition. A “high capacity magazine” does not include an attached tubular device to accept, and capable of operating only with, .22 caliber rimfire ammunition

Can you have 30 round magazines in Chicago then?

Yes. As long as you have a valid FOID and/or CCL and the magazines are for a pistol.
'High capacity' rifle magazines are not legal, but as far as we have been able to discern no FOID or CCL holder has been convicted of violating 8-20-085 in regards to high capacity magazines. However, not all
such convictions would be public record.
The language of 'large or high capacity' has been challenged in court though.

"Assault Weapon Bans" and magazine capacity:

When the FOID Act was amended along with the passage of FCCL (Illinois Concealed Carry) in 2013, the state left a window open for municipalities to enact “Assault Weapons Bans.”  Beyond that window, now long closed, regulation of long guns is a State issue:

FOID Act

Sec. 13.1

(c) Notwithstanding subsection (a) of this Section, the regulation of the possession or ownership of assault weapons are exclusive powers and functions of this State. Any ordinance or regulation, or portion of that ordinance or regulation, that purports to regulate the possession or ownership of assault weapons in a manner that is inconsistent with this Act, shall be invalid unless the ordinance or regulation is enacted on, before, or within 10 days after the effective date of this amendatory Act of the 98th General Assembly. Any ordinance or regulation described in this subsection (c) enacted more than 10 days after the effective date of this amendatory Act of the 98th General Assembly is invalid

Local Illinois municipal ordinances that regulate magazine capacity:

There are 52 Illinois cities or municipalities that enacted regulations during this "Assault Weapons Ban" window, and of those 52 only 13 of them mention magazine limits or capacity restrictions at all.

The following 13 Illinois municipalities ban large capacity magazines broadly:

Aurora| Sec. 29-49. (h)(4) Large capacity ammunition feeding device means: a. Any magazine, belt, drum, feed strip or similar device that has a capacity of, or can be readily restored or converted to accept, more than fifteen (15) rounds of ammunition;...

Deerfield| Sec. 15-86. Definitions: Large capacity magazine means any ammunition feeding device with the capacity to accept more than ten rounds...

Hazel Crest| Sec. 20-82 (5) ''Large capacity magazine" means any ammunition feeding device with the capacity to accept more than ten rounds,...

Highland Park| 134.010 (3) "Large Capacity Magazine" means any ammunition feeding device with the capacity to accept more than ten rounds,...

Homewood| Sec. 70-219 (a)(4) "Large capacity magazine" means any ammunition feeding device with the capacity to accept more than ten (10) fifteen (15) rounds,...

Markham| 137-20 (A)(5) "Large capacity magazine" means any ammunition feeding device with the capacity to accept more than ten rounds,...

Melrose Park| They did not define within the new amendment, but instead relied on their previous definition: 9.24.06 "Large capacity magazine" means any ammunition feeding device with the capacity to accept more than ten (10) rounds...

Oak Park| 27-1-1 Assault Ammunition Feeding Device: Any magazine, belt, drum, feed strip or similar device that has a capacity of, or can be readily restored or converted to accept, more than ten (10) rounds of ammunition;...

Orland Park 7-3-7-7 (3) (A) "Large capacity ammunition feeding device" means: (i) a magazine, belt, drum, feed strip, or similar device that has a capacity of, or that can be readily restored or converted to accept, more than 10 rounds of ammunition;...

Riverdale| 5.120.180 (B). "Assault ammunition" means a detachable magazine box capacity of which is more than 35 rounds centerfire...

Rosemont| Sec.15-130. Definitions: Large capacity magazine means any ammunition feeding device with the capacity to accept more than ten rounds...

University Park| Sec. 670-05. Definitions: "Large capacity magazine" means any ammunition feeding device with the capacity to accept more than ten rounds...

Winnetka| Sec. 9.12.020 (H) "Large capacity magazine" means any ammunition feeding device with the capacity to accept more than ten rounds...

NO municipal ordinances distinguish between handgun and rifle magazines specifically. 
This is important because:

Using large or high capacity magazine as a catch all term has been argued in court to be invalid because it applies to handguns as well as rifles, and handguns are subject only to state regulation. 

The question presented in the Deerfield Assault Weapon Ban challenge was whether Deerfield’s ban of large capacity magazines improperly regulated handgun ammunition. Deerfield defined “large capacity magazine” as “any ammunition feeding device with the capacity to accept more than ten rounds." 

The second district appellate court eventually threw out the magazine limit that went along with the assault weapon ban, page 33:

"We thus hold that, to the extent that Deerfield’s ban of large capacity magazines regulates ammunition for handguns, it is preempted in its application to holders of valid FOID cards and - 33 - 2020 IL App (2d) 190879 concealed carry licenses by section 13.1(b) of the FOID Card Act and section 90 of the Concealed Carry Act. Accordingly, on this limited point, we affirm the trial court’s grant of summary judgment in favor of Easterday and Guns Save Life."

Deerfield won on their assault weapon ban, but they lost on high capacity magazines.

A brief window for municipalities to establish “Assault Weapon Bans" does not permit these municipalities to regulate anything in relation to handguns or pistols.  It is important to note that this ruling applies only to the Deerfield magazine limit and only in the 2nd District.

Rulings from different districts are considered persuasive, but not binding, in other districts. Meaning the Deerfield magazine limit being struck down was good news for other districts, but it did not set precedent that can be applied outside the 2nd district.

Conclusions: 

• Local magazine capacity limits are preempted by the FOID and FCCL acts when those magazines are purchased or possessed by a FOID or Concealed Carry holder and used in a handgun.
• Rifle magazine limits were rejected in the 2nd District Appellate Court in the Deerfield case for similar reasons: Municipalities may not regulate ammunition for handguns, and this includes magazines. 
• Even though it may be perfectly legal for you to possess high capacity pistol mags under State law, you might still be arrested for violating the invalid municipal ordinances as police officers are unlikely to be familiar with the nuances of this topic. Basically, you'll beat the rap but you may not beat the ride. 
• AR or AK pistol magazines and magazines for pistol caliber carbines are more ambiguous:  Is that glock magazine for my Glock 17 at home in my safe or for the Ruger Carbine in my trunk?  Who knows?