I'm fairly certain I've seen this in the past and I highly doubt I'm making this up - I'm not sure whether it was of the British Empire or Japan's or maybe even some other nation - but I'm fairly certain I once saw a map in which one of the aforementioned nations made their own borders a whole lot bigger than they actually were as a sort of propaganda. Or it might've been an inflated size "competing" with a rival/opposing nation. I was hoping someone could confirm this.
There are a number of possibilities here, though none fully meet your criteria. Aside from the Mercator distortion mentioned by SJuan76 in his comment, you may be thinking of the McDonald Gill Highways of Empire map from 1927 which
placed the British Isles in the centre of the frame and projected, in red, the overseas empire around them, in somewhat distorted and exaggerated size.
Source: John M. MacKenzie, John MacDonald MacKenzie (eds), 'Imperialism and Popular Culture'
Note also that French Africa has been compressed, though this may not have been a deliberate attempt to belittle a colonial rival.
Source: For God and Empire 2.0
According to this British Library article, this map was
a 20ft x 10ft hemispherical projection of the world. Produced in 16 sheets, this was the largest poster ever produced at that time and required specially-built hoardings.
It was commissioned by the newly-formed Empire Marketing Board (EMB) to bolster the links and promote trade with the colonies, who were beginning to push for independence. With Britain at its centre, the map depicts the extraordinary extent of the British territories and their maritime links, with modern steamships plying the waves alongside medieval-style sea monsters.
Perhaps the best examples of exploiting the distortion created by the Mercator projection are maps such as the one below, which also includes Siam / Thailand in 'British Indochina'.
(Click on image to enlarge). Map source
It was a map such as this on which Clement Attlee, future British Labour Prime Minister commented:
On the wall at school hung a great map with large portions of it coloured red. It was an intoxicating vision for a small boy....We believed in our great imperial vision.
Cited in James R. Akerman (ed), 'The Imperial Map: Cartography and the Mastery of Empire'
An even cruder distortion of reality is the map below from 1900 which has Australia - and then quite a bit more Australia...
Image source: goldiproductions
a Canadian secondary school teacher who was a self-proclaimed publicist for the Empire...
...Parkin lobbied for the adoption of his map, and it was accepted by the 1894 Colonial Conference in Ottawa as the official cartographical representation of the Empire.
Source: Arthur Jay Klinghoffer, 'The Power of Projections: How Maps Reflect Global Politics and History'
Klinghoffer also mentions
The “Kapp globe,” made in Nuremberg about 1871–80, was produced for the British market, and it intentionally portrayed a British world image. The only country labeled in a Europe crowded with cities and rivers was “British Isles,” and its size was enhanced.
Mark S. Monmonier's How to Lie with Maps is a virtual handbook on how to get maps to deliver a particular message. Examples of expanding borders include (inevitably) maps from India and Pakistan over Jammu and Kashmir, as well as Argentine postage stamps with the Falklands / Malvinas (as cited by SJuan76 in his comment).
Often, though, the aim was to encourage sympathy for a supposedly put-upon nation; this involved emphasizing the size disadvantage of one territory over another. This tactic was used, for example, by the Nazis in Facts in Review, aimed at eliciting sympathy for Germany by (for example) comparing the size of Germany to that of the British empire.