= Solution
A <spacetime> is <geodesically complete> if every <maximal geodesic> has an <affine parameter> ranging over all of $\mathbb R$. For <timelike geodesics> this is equivalent to unbounded <proper time> in both directions. An <extendible geodesic> segment can be prolonged in the same <spacetime>; an <inextendible geodesic> cannot. A finite coordinate endpoint need not imply finite <affine parameter>.
For the <Kruskal spacetime>, use $U,V$ with
$$
UV=\left(1-\frac r{2M}\right)e^{r/(2M)},\qquad ds^2=-\frac{32M^3}{r}e^{-r/(2M)}dU\,dV+r^2d\Omega^2.
$$
In the right exterior $U<0<V$. A truncated ray $U=U_0<0$, $V_1<V<V_2$ is an <extendible geodesic> of radial null type: neither artificial endpoint is a <spacetime> boundary. A future ray $U=U_0>0$ in the <black hole> reaches $UV=1$, hence $r=0$, and is <inextendible geodesic> and future <null-geodesically incomplete>. Its <Killing energy> $E\ne0$ gives $dr/d\lambda=-|E|$, so the <Schwarzschild singularity> occurs at finite <affine parameter>. The maximal continuation toward the past supplies the other half of this same <null geodesic>.
\b[There is no inextendible, complete radial timelike geodesic in positive-mass Kruskal spacetime.] This requested example is impossible as printed. For a radial <timelike geodesic>, the conserved <Killing energy> and normalization give
$$
\dot r^2=E^2-1+\frac{2M}{r},\qquad \ddot r=-\frac M{r^2}.
$$
If $E^2<1$, $r$ has at most one turning point, a maximum $r_{\max}=2M/(1-E^2)$; a maximal trajectory runs from the <white hole> <Schwarzschild singularity> to the <Schwarzschild singularity>. If $E^2\ge1$, there is no finite turning point; one end can lie at infinity but the other reaches $r=0$. The exceptional $E=0$ trajectory through the <bifurcation surface> also reaches $r=0$ in both time directions. Near $r=0$,
$$
d\tau\sim\sqrt{\frac r{2M}}\,dr,
$$
whose integral is finite. Constant-$r$ radial <timelike curves> are accelerated, not <geodesics>.
Two plausible repairs have different meanings. Removing “radial” permits a complete circular <timelike geodesic> at $r=R>6M$, with nonzero <angular momentum> and <proper time> ranging over $\mathbb R$. Replacing “timelike” by “null” permits a complete horizon <null geodesic>: $U=0$, $V\in\mathbb R$, with $V$ an <affine parameter>. The <Penrose diagram> shows both repairs explicitly, together with the two valid requested examples; the circular trajectory is only a radial projection and is labelled as nonradial.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-311-kruskal.png]
{title=Kruskal causal diagram with extendible and incomplete null rays and explicitly labelled repairs to the impossible radial timelike example}
The horizontal boundaries are the <Schwarzschild singularities>, diagonal dashed lines are the <Killing horizons>, and outer diagonal edges are <null infinity>.
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