4026 + 7-segment display: do I need transistors?

Life can be tough. CMOS ICs from the CD4000 series, like the CD4026 can only source and sink very little current, typically 1 mA at 5 V and that will be much too little for a typical 7-segments display. So I'm afraid you'll need the transistors, especially since you're now already asking for more brightness.

But You don't necessarily need all these discrete components. A ULN2803 replaces 8 transistors, so you'll need 1 IC per display. Important note: I just read that your displays are common cathode. The ULN contains an array of NPN transistors, switching to ground, so they can only be used with common anode displays.

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starblue found a high side version of the ULN2803 in the UDN2981, so this is suitable for common cathode displays. I don't know about availability; Digikey lists only two versions, both from Allegro, as as non-stock, call for price.
Inputs are active-high, so it can be driven by the same devices which would control NPN transistors or the ULN2803.
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Alternatively you may use low-power Schottky ICs, but then you'll need the counter and the display decoder separately; I don't think the combination like the 4026 exists in LS-TTL. The 74LS90 is a decimal counter, and the 74LS247 a 7-segments decoder, which can drive LEDs directly. This is also an active low output IC, which means common anode:-(.

Other solutions you've seen requiring less transistors may have used a microcontroller and a multiplexed display. Then for 6 digits you'd only need 6 + 7 transistors, instead of 6 \$\times\$ 7.

Even if you don't want to use a microcontroller, building your own sequential-state controller might be more informative than building a board with discrete counters in addition to using less circuitry, especially if you use a multiplexed display. The guts of the controller would be a ROM (EPROM or flash) chip, a few latches, and a CD4517 (or equivalent) shift register. The display drivers would be controlled by a of 74HC595 or equivalent to drive the segments, and a 74HC138 to select a common wire. The amount of circuitry required would be affected by how efficient you wanted to be in terms of "cycle utilization" and "code-space" efficiency (e.g. how big a flash do you want to use). Using such an approach, it would be possible to design a device which would behave like a typical multi-function wristwatch (hour/minute/second, day/month/year, stopwatch, alarm, etc.) with a fairly reasonable amount of circuitry.

Edit A minimal (from a component-count standpoint) implementation which would probably work would be to use a 128Kx8 ROM, a two six-bit edge-triggered latches, two 74HC595 for display segment/common control, a '165 to read buttons, a CD4517 for data storage, and an inverter to trigger half the stuff on rising clock edges and half on falling clock edges. Probably about 15 transistors to drive the display (multilplexed with seven segments and seven commons), and miscellaneous resistors for the display driving and for button pull-ups. I would guess that, clocked at 1MHz or so, such a device could emulate a typical common 'stopwatch' chip, or do many other things, based upon the "program" in its ROM. Ask if you'd like more detail.