I do not think it is up to us to determine what is considered an Illegal Technique - TLG (The LEGO Group) determines this for their commercial products. It depends on how much of a purist you are whether you apply these constraints in your own builds.
That said, LEGO Designer Jamie Berard has made a presentation in the past that highlights most Illegal ...
This answer is based upon the wrong assumption that Technic chain links and conveyor-belt links (a.k.a. bulldozer chain links) connect to nothing except each other.
As remarked by Ivan Sanchez, this is contradicted by this set, which uses a connection between a conveyor-belt link and a standard arm/rod.
Assuming that these chain links only connect to each ...
In this case, knowing that the model is 43 cm long and 24 cm wide isn't particularly information. It's very difficult to give a rough average of parts when all the parts vary in size and the model isn't in a single primitive form (cube, sphere, pyramid).
With the exception of the eyes and mouth, this particular model appears to be built almost perfectly ...
I have another case for 4 connections.
18592 - Propeller 3 Blade 13 Diameter (Ninjago Airjitzu Flyer Vortex Rotor), can be attached to 2 x 4 brick with a help of following 3 parts below:
18590 - Technic, Gear 8 Tooth with Pin Holes and Ninjago
Flywheel Socket - Short Shaft
18585 - Brick, Modified 2 x 4 with Pin Holes and
Flywheel Socket (Ninjago Airjitzu ...
Thornax, 7 connections.
This is just a start, but here's a 2x4 connection to a BIONICLE thornax.
BIONICLE Ball, 6 connections
The extra two pieces are needed to hold the ball in place, making it truly connected.
The Nexo Knights sets have a new piece, the Bar 1L with Towball, which might work for putting a bunch of mixel joints in a small space:
Also, if you could get your hands on the maxifig hand, it's great for connecting to mixel joints. It's a 1x1 round tile with a hole in the stud and a ball on the side:
Either of these attached to the brick with 4 studs on ...
After some experimenting, here is another solution that might fit your needs. It consists of 3 alternatives:
1x 4733 Brick, Modified 1x1 with Studs on Four Sides
4x 6628 Technic, Pin with Friction Ridges Lengthwise and Towball
4x 4697b Pneumatic T Piece New Style (T Bar) (Build 1&2) OR 4x 4070 Brick, Modified 1 x 1 with Headlight (Build 3)
Since you do not give a desired size of the palm, I'd like to propose the following solution, consisting of 1x 6222 - Brick, Round 4 x 4 with 4 Side Pin Holes and Center Axle Hole and 4x 6628 - Technic, Pin with Friction Ridges Lengthwise and Towball, arranged like so:
You might need to work on the size of the fingers to make this work with "grip" - but it ...
Couple of observations:
I do not understand your teacher's motivations in giving out this exercise. There is too much guesswork involved to make the exercise of any educational value. Example: there is no visibility into the internal frame of the main "stem" of the cactus, which is clearly a SNOT build made of several layers of plates mounted on an (...