What is a Basic Function? Surely z2+c the Mandelbrot Set is a basic function.
For transcendental functions exp, log, sin, cos, tan I have tried to find the shortest possible combination with z.
In the table scaled 0.8 is the scale value of the function: scale*f(z,c): 0.8*f(z,c).
In the table s=0.25 is the start value 0.25 of the iteration. Please see math basics.
In the table the link "scaling" is a link to show the scaled variants of f(z,c).
Clicking on the thumbnail image, a large version is shown.

exp-z

scaled 1.1
s=0.0
scaling
log-z

scaled 1.1
s=1.0
scaling
zpz

scaled 3.0
s=0.367879441
scaling
sin

scaled 2.2
s=1.570796327
scaling
cos

scaled 2.0
s=0.0
scaling
z*tan

scaled 1.3
s=0.0
scaling
\(\mathbf{\bf\:exp(z) - z + c}\)
\(\mathbf{\bf\:log(z) - z + c}\)
\(\mathbf{\bf\:z^z + c}\)
\(\mathbf{\bf\:sin(z) + c}\)
\(\mathbf{\bf\:cos(z) + c}\)
\(\mathbf{\bf\:z*tan(z) + c}\)
z*asin

scaled 1.0
s=0.0
scaling
z*acos

scaled 1.0
s=0.652180
scaling
z*atan

scaled 0.7
s=0.0
scaling
sec

scaled 2.4
s=0.0
scaling
csc

scaled 2.0
s=0.0
scaling
z*cot

scaled 1.0
s=0.0
scaling
\(\mathbf{\bf\:z*asin(z) + c}\)
\(\mathbf{\bf\:z*acos(z) + c}\)
\(\mathbf{\bf\:z*atan(z) + c}\)
\(\mathbf{\bf\:sec(z) + c}\)
\(\mathbf{\bf\:cosec(z) + c}\)
\(\mathbf{\bf\:z*cot(z) + c}\)
z*asec

scaled 0.6
s=-1.061930
scaling
acsc + z

scaled 1.0
s=1.27201965
scaling
z*acot(z^2)

scaled 4.465
s=0.847657
scaling
z*sinh

scaled 1.0
s=0.0
scaling
cosh

scaled 1.3
s=0.0
scaling
z*tanh

scaled1.5
s=0.0
scaling
\(\mathbf{\bf\:z*asec(z) + c}\)
\(\mathbf{\bf\:acsc(z) + z + c}\)
\(\mathbf{\bf\:z*acot(z*z) + c}\)
\(\mathbf{\bf\:z*sinh(z) + c}\)
\(\mathbf{\bf\:cosh(z) + c}\)
\(\mathbf{\bf\:z*tanh(z) + c}\)