How to produce a binary output sequence using LFSRs but with increased linear complexity?
One standard way is as follows. Given $n$ primitive LFSRs with lengths $L_i,i=1,\ldots,n$ such that $\gcd(L_i,L_j)=1,$ and a nonlinear boolean function $$f:\mathbb{F}_2^n \rightarrow \mathbb{F}_2$$ with $f(x_1,\ldots,x_n)$
you have output linear complexity
$$
f(L_1,\ldots,L_n)
$$
evaluated over the integers, where output of LFSR $L_i$ is input as the variable $x_i$ of the boolean function. This is called the combination generator.

So if, for example,
$$f(x_1,x_2,x_3)=x_1x_2+x_2x_3
$$
with
$L_1=31,L_2=32,L_3=33,$ the linear comlexity becomes $31\times 32+32\times 33=2048 $.

See the book Encyclopedia of Cryptography and Security on page 233 for a good discussion.