What’s an intuitive way to understand Taylor and Maclaurin series?

I don't know what you expect exactly, there is the "Why do we have all derivations" and "Why do we have 1/n!". The first is intuitive, he second is not, so here is some ugly computation.

You have a car. Position x, speed v, acceleration a.
Your car goes at v(0) km/h (constant speed). It begins at x(0), what is x(t) ?
Obviously :
x(t) = x(0) + v(0)*t

Or more precisely :
x(t)
= x(0)    +    int[0->t]( v(t) * dt)
= x(0)    +    int[0->t]( v(0) * dt )
= x(0)    +    v(0) * int[0->t](dt)
= x(0)    +    v(0) * t

But dammit, my car was not moving at constant speed ! v(t) is not constant, there is a constant acceleration a(0).

Let's try again :
Obviously :
v(t) = v(0) + a(0) * t =  v(0) + int[0->t]( a(0) * dt)
so
x(t)
= x(0)    +    int[0->t]( v(t) * dt)
= x(0)    +    int[0->t](  v(0) * dt + int[0->t]( a(0) * dt) * dt )
= x(0)    +    v(0) * int[0->t](dt)    +    a(0) * int[0->t]( int[0->t](dt) * dt )
= x(0)    +    v(0) * t   +   a(0) * t²/2

But dammit, my car was not moving at constant acceleration ! a(t) is not constant, there is a x'''(0) or v''(0) or a'(0).

And we do it again and again, until we are bored. And by adding each time a correction, we have a serie. The 1/n! comes from the serial integrations.
dt -> t
t * dt -> 1/2 * t²
1/2 * t² * dt -> 1/6 * t³
And so on.

The serious math will explain you why it has a limit.
You have the pure "infinite" serie, and tons of "partial" sums with a way of computing the error (Taylor – XXX formulas).

What's an intuitive way to understand Taylor and Maclaurin series?