Recall that a mouse is a structure of the form $(J\_\alpha[U],\in,U)$ with $U$ being an amenable ultrafilter with some iterability properties.


One of the interesting facts about mice is that given two mice, $m$ and $n$, there is an ordinal $\lambda$, such that if we iterate both mice $\lambda$ times, one is going to be an initial segment of the other.


This induces a pre-well ordering on the mice of the universe. In a recent course, we saw that there is no maximal element to this order. If you have a mouse, then you can define a slightly larger mouse. This means that if we look at $L[0^\#]$, which is a model with one nontrivial mouse, then there are infinitely nontrivial mice there, which are strictly larger than $0^\#$ itself.


Assume that in $V$ the mice pre-well ordering has order type of $\rm Ord$, and let $M$ be an inner model such that the mice in $M$ are only those below $\alpha$ in said order, and $m$ is the least mouse not in $M$.



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> What can we say about the ordering of the mice of $M[m]$? How many new non-equivalent mice have been added? In particular, what is the mice ordering in $L[0^\#]$?
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Clearly there order type is at least $\alpha+\omega$. But it's probably much more. How much more?



How verminous are mice?
Looking at your example of mouse, it seems you are going to use, and are asking about, the ‘old-fashioned’ fine-structural mouse, as used originally by Dodd and Jensen in their papers, and by Dodd in his book. It is crucial to say this since the ‘modern’ definition involved a reorganisation of these hierarchies following a suggestion of S. Baldwin in the 1980’s. The ‘modern’ organisation is the current approach to mice in all the various versions of Jensen and Mitchell-Steel. (It essentially ensures that a strong form of GCH applies level-by-level, which the Dodd-Jensen approach did not. This property is called (strong) acceptability.)


The upshot is that Dodd-Jensen have mice but a special subclass of these are the ’critical mice’ (and $O^\sharp$ is the first such, $O^{\sharp\sharp}$ is the second such, &c.). A critical mouse engenders an inner model by iterating it’s measure out through the ordinals. A non-critical DJ-mouse does not. (This is a major difference with modern mice: iterating a modern mouse by its topmost active extender repeatedly will always leave behind an inner model.)


So mouse from now on means a Dodd-Jensen mouse, and $K$ will be the Diodd-Jensen core model built below the first inner model with a measurable cardinal.


One may construct (non-critical) mice from critical mice, so that, as you say in $L[O^\sharp]$ then there are many mice definable from $O^\sharp$. You mention infinitely many but in fact there are a proper class many of such. So the length of the mouse order in $L[O^\sharp]$ is $On$ itself. In $L[O^{\sharp\sharp}]$ it is $On + On$ and so on. 


More manageably: in any model or V, we may consider the mice in HC, or in $H\_\kappa$ for some regular $\kappa$. These can all be iterated to a common hierarchy generated by the c.u.b. filter on $\kappa$, $F=F\_\kappa$ say. This common iterate is of the form $J\_\theta^F$ and is the direct limit then of all the mice in $H\_\kappa$. The ‘strength’ of the mouse ordering can then be gauged by properties of $\theta$, (if it admissible, if it is . . .) but always $cf(\theta) = \kappa$ in any universe. (As an example, in $L[O^\sharp]$, thinking of the mice in $H\_{\omega\_1}$ they iterate to $J^F\_{\omega\_1 + \omega\_1}$ where $F$ is the c.u.b. filter on $\omega\_1$. In $L[O^{\sharp\sharp}]$, $\theta$ would be $\omega\_1 \cdot 3$.)


Thus, to answer the question specifically, the mouse ordering in $L[O^\sharp]$ of the mice in $H\_\kappa$ has length $\kappa$. In $L[O^{\sharp\sharp}]$ it is $\kappa\cdot 2$ *&c.* In the least model closed under $\sharp$’s it will have length $\kappa\cdot\kappa$. Once the model becomes even a little bit thicker there is no handy description of the length of the ordering in terms of some ordinal arithmetic. (Some of this is summarized in Sect. 3
of my *“On unfoldable cardinals, $\omega$-closed cardinals and the beginnings of the inner model hierarchy”*, Archive for Math. Logic, vol. 43, No. 4, (2004), pp.443-458.)


The common iterate $(J^F\_\theta, \in F)$ for $F=F\_\kappa, \theta = \theta(\kappa)$ Dodd and Jensen called the “$Q$-structure at $\kappa$” - thus $Q\_\kappa$ (but again this has a totally different usage in modern mouse theory). Once the $Q$ structure is admissible interesting things can happen. As a sample (see again cited paper):


Theorem (V=K) If there exists a $\kappa$ with $Q\_\kappa$ admissible, then 
$\forall D \subseteq \omega\_1( D$ is universally Baire $\leftrightarrow \exists r\subseteq \omega (D \in L[r])$.


Theorem (V=K) If $Q\_{\omega\_1}$ is admissible, then $\delta^1\_2=\theta\_{\omega\_1}.$


These results are not lost by converting to the modern mice concepts. There are reasonably natural analogues. And to emphasise, the Dodd-Jensen $K$ has exactly the same sets as the 'modern' $K$ below a measurable cardinal. Most simply by forgetting about the non-critical mice in the Dodd-Jensen mouse ordering you will get an ordering of the same order type as the modern one of 'active' mice (or equivalently the order type of the inner models of $K$ under comparison).


