Obviously this would be a consistantly high(above sea level) country but which one has the highest ?
WAG: Nepal.
According to the CIA World Factbook, Rwanda has the highest low point at 950 meters (3117 ft) above sea level. Other notables are Andorra at 840 meters (2100 ft) and Armenia at 400 meters (1000 ft).
http://www.odci.gov/cia/publications/factbook/fields/elevation_extremes.html
And since Chas snuck in there whilst I was typing my first reply, Nepal’s low point is a mere 70 meters (175 ft).
Thank you.
Good site. 
Interesting page, but I noticed this:
Lesotho:
lowest point: junction of the Orange and Makhaleng Rivers 1,400 m
And hey, now that I think about it, how the HELL can the lowest point in a country be a river junction? Water doesn’t flow uphill!
That river confluence is at the border of Lesotho. Downriver from there is outside the country.
Is that river junction at the border?
<sigh>
This map and this map show that these rivers intersect at the border, thus it makes sense for this to be the low point in the country.
Now, if yojimbo had asked the question the other way 'round, I’m sure my fair country The Netherlands would have made the Top Ten with our mightily intimidating Vaalserberg, with a whopping 321 metres above sea level.
Good catch, Chas. Thanks for correcting me.
Coldie, I’d be willing to be there’s a bunch of tiny little island nations that have lower high-points than your country.
Nauru:
lowest point: Pacific Ocean 0 m
highest point: unnamed location along plateau rim 61 m
Navassa Island:
lowest point: Caribbean Sea 0 m
highest point: unnamed location on southwest side 77 m
Yeah, I noticed those little buggers too, Unc. I even found one that had a highpoint of 1.8 metres. Sheesh, talk about showing off.
A nice little tidbit, also from your linked site: the Netherlands is slightly smaller than two times New Jersey. Remember that the next time you read we have 16 million inhabitants.
I think I remember the Maldives being in the Guinness Book of Records for the lowest highest point. I would have known Lesotho as well, if I had seen the OP on time! Anyway, the Netherlands is not even the lowest country in Europe, as far as I know. I think Denmark is even lower.
The highest peak in the Danish Alps, Yding Skovhøj, measures a staggering 173 metres above sea level
I’m not sure this is possible. At least in Ireland we measure altitude from an Ordnance Datum which refers to spring low water on a certain date. An island whose highest point was 1.8 m above low water would be maybe 2 to 5 m below sea level at high tide.
Most countries use “mean sea level,” which is the mean of mean high tide and mean low tide. This site, says the UK does it differently from the way you describe it for Ireland.
Tides differ globally too, and are especially affected by coastal configurations. I think most oceanic islands have fairly small tides.
According to the Fact Book linked to above, the Maldive Islands in the Indian Ocean seems to be the country with the lowest high point, 2.4 m, although a few island possessions like Howland Island are as low as 1 m. This indicates the tides there must be minimal.
The highest point in Denmark is the top of one of the towers supporting the bridge across Storebælten 
And now that I think about it, this is still a good question. I mean, those rivers must be contained in something. Wouldn’t the riverbank necessarily be higher than the river itself?