As we know, when the Poisson rate $\lambda$ is large, we can approximate the Poisson distribution as a normal distribution with equal mean and variance.
Is there a conjugate family for this normal distribution with equal mean and variance? Neither gamma nor inverse-gamma work for that.
Let $X_1, X_2, \dotsc, X_n$ be iid from the distribution $\text{N}(\theta,\sigma^2=\theta)$, that is, a normal distribution with equal expectation and variance ($\theta>0$). There is a conjugate distribution for this model, which is known as generalized inverse gaussian distribution, see https://en.wikipedia.org/wiki/Generalized_inverse_Gaussian_distribution#Conjugate_prior_for_Gaussian
The likelihood from the normal model above can be written (after dropping constants) as ( * ) $$ L(\theta) \propto \theta^{-n/2} e^{-(n/2)\theta} e^{-\frac{\sum_i x_i^2/2}{\theta}}, \qquad \theta>0 $$ so we need a prior distribution with three factors that can absorb the three factors from the likelihood. We propose then $$ \pi(\theta) =K^{-1} \theta^{-a-1} e^{-b/\theta} e^{-c\theta} $$ with $a\in\mathbb{R}, b>0, c>0$ and $$K={\frac {{c}^{a/2-1}{b}^{a/2}{c}^{-a/2} \left( -{b}^{-a/2}a{{\it K}_{a} \left(2\,\sqrt {cb}\right)}+{b}^{1/2-a/2}\sqrt {c}{{\it K}_{a+1}\left( 2\,\sqrt {cb}\right)} \right) }{{{\it K}_{a}\left(2\,\sqrt {cb}\right) }}} $$, $K$ is the modified Bessel function of the second kind.
Then we can find the posterior as $$ \pi(\theta | x) \propto \theta^{-(n/2+a)-1} e^{-(n/2+c)\theta} e^{-(b+\sum_i x_i^2/2)/\theta} $$
( * ) I think this model is an interesting example. Note from the likelihood factorization above that $T=\sum_i X_i^2$ is a sufficient statistic. In the question the model was developed as an approximation for a poisson model, but in the poisson model the sufficient statistic is $T^* = \sum_i X_i$. So, however good the approximation is, the two models lead to different sufficient statistics!
