Different countries will have different requirements in terms of paperwork and for mountaineering expeditions these frequently extend beyond a simple travel visa. This page gives an overview of some of the more common bureaucratic requirements for climbing in the greater ranges.
Visas
Depending on where you go there will inevitably be bureaucracy of some kind. South America is unique in that visas are not usually required for British nationals and there is no formal requirement to register what you plan to do. This appears to be generally true for Alaska also, although for entry into any of the US states a valid ESTA, available on-line on demand for a modest sum, is essential before you leave the UK. For most Asian countries a visa will be required. These are usually obtained in advance from either the relevant embassy or High Commission or, more commonly nowadays, through an official visa service.
Do NOT be tempted to use non-authorised visa services as these invariably charge more and are not necessarily any quicker. It is important to apply for visas in good time as not doing so can result in the application not being processed in time for your departure from the UK. At the present time India is likely to request a specific mountaineering visa (mountaineering or X-visa), and NOT a tourist visa, if you intend climbing in specified areas or peaks in excess of 6,000m. Visas of this type can only be obtained following receipt of a specific request from the Indian Mountaineering Foundation which is usually only available a short while before the expedition is due to leave the UK, so timing here is critical.
If you wish to visit Russia and some of the former Russian states, you will require an invitational letter before you can even apply for a visa. Find out well in advance whether or not this is obligatory for the country that you wish to visit. If it is, then the in-country agent that you use to help with logistics should be able to provide it. Such invitations will require all of the expedition participants to be named in advance.
Permits
If you plan to visit an area covered by a National Park in India you will be expected to possess the relevant permit and to have paid the daily rate for that park. Unfortunately foreigners pay considerably more than Indian Nationals but this cannot be avoided. Such permits are usually organised in advance by your local in-country agent. For higher peaks and sensitive areas, the Indian Mountaineering Foundation will demand a peak fee payable in advance by your agent (and effectively by your expedition). Such fees are set on a fixed scale and increment in 500m intervals from 5,500m and take into account the number of members in the party. In these instances after arriving in Delhi you will be required to visit the IMF offices for a briefing, to sign certain declarations and to meet your allocated Liaison Officer (LO), for whom you will have hired gear from the IMF (in 2012 the cost for this was $500). The LO will accompany the party to base camp and may climb with you or stay at base camp until the return to Delhi. Generally the LOs are a help rather than hindrance but in addition to facilitating your expedition they are required to monitor your activities. A second visit to the IMF will be required before you depart for the UK.
The bureaucracy in most other countries is usually less complex than in India but in the Caucasus there are current restrictions on climbing on the border with Georgia. This eliminates climbing on mountains whose summits span the frontier.
Additional Costs
The best way to find out information on luggage shipping and staff costs is by speaking to another recent expedition to the same area. They will have the most up-to-date information and may even be able to recommend an operator or agent.