Remove["Global`*"]

Series expansion of exact function:

Series[LambertW[T],{T,0,10}]

Initial approximation compared to exact function:

n=3;Series[LambertW[T]-(T/(1+T)),{T,0,n}]

Second order polynomial approximation compared to exact function:

Solve[c==1-c T+(((c^2)(T^2))/2),c]

n=4;Series[LambertW[T]-((1+T-(Sqrt[1+2 T-(T^2)]))/T),{T,0,n}]

Third order polynomial approximation compared to exact function:

Solve[c==1-c T+(((c^2)(T^2))/2)-(((c^3)(T^3))/(3!)),c];

n=5;Series[LambertW[T]-T((1/T)-((18 (T^3)+9 (T^4))/(9 (T^3) (((-3 (T^5)+(T^6)+(Sqrt[8 (T^9)+21 (T^10)+2 (T^12)])))^(1/3))))+((((-3 (T^5)+(T^6)+(Sqrt[8 (T^9)+21 (T^10)+2 (T^12)])))^(1/3))/(T^3))),{T,0,n}]

Fourth order polynomial approximation compared to exact function:

Solve[c==1-c T+(((c^2)(T^2))/2)-(((c^3)(T^3))/(3!))+(((c^4)(T^4))/(4!)),c];

n=6;Series[LambertW[T]-T((1/T)-(1/2) √(-(4/(T^2))+((4 (2^(1/3)) (-2 (T^3)+(T^4)))/((T^4) (((9 (T^4)+12 (T^5)-(T^6)-(Sqrt[81 (T^8)+248 (T^9)+78 (T^10)-3 (T^12)])))^(1/3))))+((2 (2^(2/3)) (((9 (T^4)+12 (T^5)-(T^6)-(Sqrt[81 (T^8)+248 (T^9)+78 (T^10)-3 (T^12)])))^(1/3)))/(T^4)))+(1/2) √(-(8/(T^2))-((4 (2^(1/3)) (-2 (T^3)+(T^4)))/((T^4) (((9 (T^4)+12 (T^5)-(T^6)-(Sqrt[81 (T^8)+248 (T^9)+78 (T^10)-3 (T^12)])))^(1/3))))-((2 (2^(2/3)) (((9 (T^4)+12 (T^5)-(T^6)-(Sqrt[81 (T^8)+248 (T^9)+78 (T^10)-3 (T^12)])))^(1/3)))/(T^4))-((-(128/(T^3))+((192 (1+T))/(T^4)))/(4 (Sqrt[-(4/(T^2))+((4 (2^(1/3)) (-2 (T^3)+(T^4)))/((T^4) (((9 (T^4)+12 (T^5)-(T^6)-(Sqrt[81 (T^8)+248 (T^9)+78 (T^10)-3 (T^12)])))^(1/3))))+((2 (2^(2/3)) (((9 (T^4)+12 (T^5)-(T^6)-(Sqrt[81 (T^8)+248 (T^9)+78 (T^10)-3 (T^12)])))^(1/3)))/(T^4))]))))),{T,0,n}]

Second order functional approximation compared to exact function:

(Subscript[C, 1])=(1/(1+T));
f[C_]:=((1+C T)/(T+Exp[C T]))
(Subscript[C, 2])=f[(Subscript[C, 1])];
n=7;Series[LambertW[T]-T (Subscript[C, 2]),{T,0,n}]

Third order functional approximation compared to exact function:

(Subscript[C, 3])=f[(Subscript[C, 2])];
n=15;Series[LambertW[T]-T (Subscript[C, 3]),{T,0,n}]

Fourth order functional approximation compared to exact function:

(Subscript[C, 4])=f[(Subscript[C, 3])];
n=31;Series[LambertW[T]-T (Subscript[C, 4]),{T,0,n}]

Fifth order functional approximation compared to exact function:

(Subscript[C, 5])=f[(Subscript[C, 4])];
n=63;Series[LambertW[T]-T (Subscript[C, 5]),{T,0,n}]

A few Padé approximants:

PadeApproximant[LambertW[T],{T,0,{1,1}}]

PadeApproximant[LambertW[T],{T,0,{2,2}}]

PadeApproximant[LambertW[T],{T,0,{3,3}}]