"Not tai chi"
Many tai chi teachers espouse physical development through means other than tai chi, and reserve tai chi for the refinement of skill. Similar to pre-war Aikido (wherein students were required to have significant expertise in other arts such as karate, judo or jiujitsu), tai chi strength and conditioning is often mixed with other styles of internal and external Chinese martial arts. Practice of the arts was traditionally porous, with students training xingyi, bagua, tai chi, and other arts either successively or simultaneously.
For instance, Hung I-Hsiang trained multiple arts:
[His] internal arts training program included xingyiquan, baguazhang and Wu (Hao)-style taijiquan, Shaolin kung fu and qigong.... Hung believed that, in practicing the xingyiquan five elements as an introduction to the internal martial arts, the student can clearly understand the way the body should be trained to move in the internal styles. If the student starts out in taijiquan it is very difficult to develop and understand internal power.
In this way, much tai chi strength training is borrowed or classified more generally. For instance, Hung's students practiced fuhugong (crouching tiger exercises). Other forms of stance work were common in Chinese martial arts, but were not specific to tai chi. However, tai chi practitioners could often be assumed to use them.
That being said, there are a few practices that we can pin on tai chi, either because they are integrated with the art directly by some teachers, or because some teachers consider the exercises to be central to developing the physical strength and power necessary for executing proper tai chi form and technique.
According to tai chi teacher Cheng Tsang Lu via Jacob Fitisemanu's article in Kung Fu magazine, tai chi strength exercises include the Chinese stone lock, which "has been part of traditional kung fu regimens since at least the Han Dynasty (206 BC-220 AD)".
The traditional stone lock workout consists of swinging and lifting routines performed in sets of several repetitions (from 5-50, depending on ability) after which the lock is transferred (often by spinning or flipping the lock in midair) to the alternate hand.
The basic lock maneuvers for building strength consist of swinging and hoisting the lock while gripping the handle. Advanced exercises include manipulating the lock with one’s elbows/forearms, spinning the stone in midair, or heaving the weight while maintaining low stance postures.
The stone lock can be used in many ways. Flipping it, flinging it in a circle, and swinging it seem to be the primary themes, as seen in those shuai jiao clips. I do not see a way that this practice would not build tremendous gripping and seizing (na) strength, in addition to hip power, shoulder and back strength. The implement seems strikingly similar to the kettlebell. (One wonders if there was any cross-pollination of the practice across Asia, and may have influenced the Russians. It is clear that the stone lock piggybacked on martial arts to Okinawa, as evidenced by its prevalence in karate (PDF).)
Heavy Round Stone
Another implement common to Chinese martial arts, including tai chi, is a big ol' rock that you lift:
Occasionally seen in movies and picture books, the “big stone ball” (da shi qiu) is a rarity in contemporary training regimens. The origin of using round boulders for bodybuilding is said to have emerged during the era of Emperor Shi Huangdi—whose civilian construction workers became renowned for their muscular strength and endurance after just a few months of hauling rock. To many people, the round stone has become affiliated with Tai Chi and Chi Kung, but its benefits transcend regional or systemic boundaries. Common exercises consist of rolling the ball at torso level (like many Tai Chi postures), coordinating “walking” routines, and slowly raising and lowering the weight in front and to the sides of the body.
This is strikingly similar to the modern strongman practice of deadlifting, carrying, and shouldering Atlas stones.
Miscellaneous resistance contraptions
Some tai chi teachers, such as "Grandmaster" Tu, practice and teach a variety of strength, flexibility, and conditioning movements. Tools include homemade suspension gear, wooden spheres, logs, gripping exercises using springs, and a kind of primitive Swiss-Family-Robinson style Smith-machine (see video for examples). While this seems to work for Tu, it is necessary to point out that the explanations he gives for how these methods work sound like complete pseudoscience.
As Eric Sbarge explains, static stance work has been a staple of tai chi training for as long as there's been tai chi:
Ch'ang Dung Sheng, renowned as the "king of Shuai Chiao" and founder of Ch'ang Style Tai chi Ch'uan, was one master who maintained stance training was a key to his martial arts success. He won China's national kung fu tournament at the age of l7 and then remained undefeated until he passed away in 1986 at age 80. His early training included holding postures such as Leaning Forward Searching for the Sea for up to an hour at a time. This particular posture allowed him to develop such sweeping and lifting power with his legs that he could actually pull small trees and saplings out of the ground by wrapping his leg around them.
Chen village instructor and push-hands champion Chen Ziqiang agrees that low-stance work is vital:
First you need a good root. To achieve this, Chen stylists practice in low stances that build strong legs. Strong legs enable you to move faster and push harder.
As he demonstrates here:
It is not clear whether he is referring to moving stances here.
"Unusual" bodyweight movements
Tim Cartmell relates the following:
I teach Zhao Bao Tai Ji Quan. Actually, I have only had one student in the seven years I have been back teaching in the States that has actually learned the entire form, it has proven too physically difficult for the rest of the students that have tried to learn it. There is a basic set of conditioning exercises that students learn at first, my teacher's standard was 36 repetitions of each exercise before learning the form.
These feats of strength that students must accomplish before learning the form include:
Twisting, Alternate Knee Drop Squats: Stand in a low 'horse' stance with the feet parallel and about shoulder width and your thighs parallel with the ground. Your arms hang at your sides and your back is straight. Twist your hips to the left and 'close' the right hip toward the left hip as you drop your right knee to touch the ground lightly (your left thigh remains parallel with the ground). As you turn left, your right shoulder also 'closes' and your right arm twists inward. Then you turn back through the center position, lifting your right knee then repeat dropping the left knee to the mat as you twist to the right. You need to stay in a full squat (thighs parallel to the ground) as you transition left to right. Work up to the minimum number of reps, 36 on each side.
Butterfly stretch squat: A more exotic looking basic exercise is to sit in the common 'butterfly' stretch position (bend your knees and put the soles of the feet together in front of your crotch). Then lift the body up so you are balanced on the outside edges of your feet only (soles of the feet pressed together). Then, keeping the soles of the feet pressed together, straighten your legs until your legs are straight (the soles of your feet must remain pressed together, you balance on the outside edges of the feet only. Be careful, there is a great stretch on the outside ankle). Squat up and down in this fashion, balancing on the outside edsges of the feet with the soles pressed together.