6. The system of claim 1 , wherein the server instructions, when executed by the data processing apparatus, further cause the data processing apparatus to perform operations comprising: accessing query log data that includes queries submitted by users and the frequency with which each query was submitted; determining most likely sequences of initial query characters from the query log data; for each most likely sequence, determining most likely queries from the most likely sequence and the query log data, each most likely query having an initial sequence of query characters matching the most likely sequence of initial query characters; and designating the most likely queries determined for each of the most likely sequences of initial query characters as the initial query suggestions.
6. The system of claim 1 , wherein the server instructions, when executed by the data processing apparatus, further cause the data processing apparatus to perform operations comprising: accessing query log data that includes queries submitted by users and the frequency with which each query was submitted; determining most likely sequences of initial query characters from the query log data; for each most likely sequence, determining most likely queries from the most likely sequence and the query log data, each most likely query having an initial sequence of query characters matching the most likely sequence of initial query characters; and designating the most likely queries determined for each of the most likely sequences of initial query characters as the initial query suggestions. 7. The system of claim 6 , wherein: the second instructions cause the client device to: provide the query characters input in the first query input field to the data processing apparatus as a query suggestion request; receive, in response to the request, additional query suggestions from the data processing apparatus; and store the additional query suggestions in the suggestion store; the server instructions cause the data processing apparatus to: receive the query characters from the client device; determine most likely sequences of subsequent query characters from the query log data, the subsequent query characters being query characters that are subsequent in sequence to the query characters received from the client device; for each most likely sequence of subsequent query characters, determining most likely additional queries from the most likely sequence of subsequent query characters and the query log data, each most likely query having an initial sequence of query characters matching the query characters received from the client device and the most likely sequence of subsequent query characters; designating the most likely additional queries determined for each of the most likely sequences of subsequent query characters as the additional query suggestions; and providing the additional query suggestions to the client device in response to the query suggestion request.