Originally none of them did, but barriers were added over the years to help ensure everyone pays the correct fare. However there remain a handful of tube station entrances where the hassle of installing barriers has never been deemed financially worthwhile. These are they.
Stations with no ticket barriers
The tube's least used station has never merited barriers. This may be because the station has exits from both platforms, upgrading one being technically pointless and upgrading both being excessively expensive. It may be because nipping across the footbridge is an important feature of local life and must not be blocked. It may be because this was one of the first staffless stations and there'd be no point in having barriers if they always had to be left open. It may be because this is a step-free station and barriers would only get in the way. Or it may just be because it's the tube's least used station, and a few faredodgers amongst the 500 passengers a day will never dent TfL's finances.
South Kenton was built as a railway station, not a tube station, so lacks some of the infrastructure which might have made barriering easier. Its island platform is isolated in the middle of the West Coast mainline, so can only be accessed up a staircase from a subway beneath the tracks. This subway is crucial to pedestrian connectivity hereabouts, so cannot be barriered, and the foot of the staircase is much too close to the subway for a row of ticket gates to have been installed. There's definitely room uptop, but never the will to add any, so passengers at the Bakerloo's least used station have to be trusted to touch in.
Mill Hill East
Mill Hill East is the least used station on the Northern line, as befits the sole outpost on a stumpy spur line. Its problem is narrow access throughout - a very small ticket hall leading directly to a narrow staircase up to the platform. Adding a couple of ticket gates isn't practical without a partial rebuild, and that's never been on anybody's agenda. Intriguingly this sets up the possibility of a barrier-free journey to the next station down the line...
Stations with one un-barriered entrance
Finchley Central station has two entrances, one to Station Road and one to Chaville Way, linked by a single covered footbridge across the tracks. Intriguingly the main entrance is on the Chavllle Way side, which is fully staffed and barriered, whereas the Station Road entrance has nothing. It was due for a Charles Holden upgrade as part of the Northern Heights plan, but WW2 scuppered that and no station building has ever appeared. Instead passengers funnel down a ramp to the mouth of the footbridge where two Oyster pads await, plus a notice warning that this entrance is for "ticket holders and disabled access only". Well used, and possibly well abused.
Two stops towards High Barnet, Woodside Park station has an extra ungated entrance on the northbound platform for the convenience of local residents.
The northbound platform has an ungated sideentrance leading to the car park and down into Chorleywood's main shopping street. A couple of card readers have replaced the Unpaid fares honesty box.
Chalfont & Latimer
The northbound platform has an extra entrance from Station Approach through a gate in a metal fence. It's labelled Ticket Holders only.
West Harrow's two platforms are accessed separately from the street on either side of the railway viaduct. The main building, such as it is, has barriers which lead to the eastbound platform. Access to the westbound looks more like an alleyway than a station entrance. A card reader is positioned halfway up the stairs.
Finsbury Park tube station finally got ticket barriers a few years ago, with one line near the Arsenal shop and another at the beginning of the subway from Seven Sisters Road. Both are a lot narrower than the optimal row of ticket barriers would be. But the National Rail entrance remains defiantly ungated, and if you head in that way it's still possible to get down to the tube platforms via (either of) the old spiral staircases, or via the newly-installed step-free lifts.
Stations with more than one un-barriered entrance
Both Waterloo & City line platforms at Waterloo are ungated, with stairs and long ramps leading into the wider warren of the mainline station. This is how it's always been, reflecting priorities long before London Underground took over. But barriers are present at the other end of the line, at Bank, should you have plans on exiting there.
Until 1982 Buckhurst Hill station had extra entrances at the southern end of each platform, after which they were sealed off for fare-collecting reasons. Last year they were reopened, with connecting ramps, to create the tube's 74th step-free station. Other than extra card readers, it's very much a cut-price solution.
Stations with one un-barriered entrance some of the time
At the far end of Pinner's northbound platform is the Cecil Park exit. It's only open on weekday mornings from 7-10am and on weekday evenings from 4.30-9pm. The rest of the time the gate across the top of the steps is locked shut, so best not touch your card on the pads alongside.
A gate at the end of the southbound platform, connecting to Wentworth Avenue, is unlocked for two hours (7.30-9.30am) during the morning peak. Those with a RADAR key can pass through at any other time.
The Hillcrest Way entrance/exit (alongside platform 1) is open from 6.30-9.30am and 4pm-8pm Monday to Friday.
You tell me that a side entrance for northbound trains is open during the evening rush hour from about 4.30pm.
Stations with a temporary un-barriered entrance
Moorgate's big Crossrail entrance isn't fully open yet, but the Circle line platforms already have an extra exit leading out to Moorfields. It's somewhat makeshift so far, and leads out to a bit of a building site, but is jolly useful for disgorging city workers. It's also closed after 7.30pm and all day at the weekends, so I wasn't able to use it yesterday, but last time it only had card readers and if that's changed I'm sure you'll tell me.