I've looked for some information on this, but couldn't find anything useful. Has there been any noteworthy attempt to estimate the sum amount of individuals of all species that have ever lived on Earth?
This is a hard problem - estimates of total living things in an given environment are usually created by looking at the number of species and individuals found in a sample area and extrapolating. As far as estimating the number of living things in the world, there are still lots of species which are not known, making this number still unknown for the world today as it is. Doing a similar survey into past ages adds another layer of problems to the estimate.
EO Wilson has, among others I'm sure, made a serious effort to understand how many species are currently alive on Earth. Its pretty clear that currently we don't know how many species are currently alive - that would make it nearly impossible to answer your question historically.
from "The barometer of Life" Stuart, Wilson, McNeely, Mittermeier & Rodriguez, emphasis mine.
As you can see there are some important technical obstacles to even the question of how many species there are. Bacteria and free living eukaryotes have rules for speciation that are fairly difficult to define, esp since many can transfer just a few genes with many near relatives and do not require sexual reproduction.
Another obstacle is that the animal and plant surveys are not really done yet either. Plants animals and insects are still being discovered in jungles, deep underwater and at rest stops. As far as the microbial record goes, an average glass of water from a pond or river probably has undocumented species.
The encyclopedia of life is a resource worth mentioning here. It is self admittedly only a partial record.
For past speciation records, the geographical sampling is not strong - digging down in selected spots where fossil records remain is bound to miss many of the species which had existed.
In 1990, Peter Dodson wrote this review in PNAS holding forth on the taxonomical difficulties in estimating the number of dinosaur species which included a founding period which was as much circus as science as well as the incredibly sparse sampling of prehistoric surveys over time and history. As recently as this year the count of dinosaurs has continued to vary wildly. A pubmed citation list to Dodson's paper shows 7 citing works, mostly from the past 3 years.
it is impossible to know the exact number so here is my gross ballpark estimate of an upper bound - i.e. the maximum number of organisms that could have lived on earth in the extreme best case scenario. in practice it is probably much less, but this is to get an idea of what kind of numbers we are dealing with.
The earth's volume is about 1.08321 * 10^12 cubic kilometers = 1.08321 * 10^21 cubic meters
The crust of the earth is 4-8km and the radius is about 6,353km giving around 1% of the total volume available for life.
Hence, the volume available, i.e. the volume of the crust is about 1.1 x 10^19 Meters.
An average-size bacterium such as the rod-shaped E-coli is about 2 micrometers long and 0.5 micrometers in diameter. Hence, the volume is roughly:
(2*10^-6)*(0.5*10^-6)*(0.5*10^-6) = 0.5*10^-18 meters
Hence, if the entire volume of the crust of the earth were entirely filled with e-coli bacteria, there would be:
(1.1 x 10^19 meters)/(0.5*10^-18 meters) = 1.3*10^36 bacteria on the earth.
Bacteria take time until they reproduce. If they reproduce faster than they grow, they will become reduced to zero size, so there must be a balance between growth and reproduction.
Under ideal conditions the E-coli bacterium would divide every twenty minutes. source
Let's say we have the maximum number of bacteria on earth dividing every twenty minutes. Somehow, they divide and then one dies so that the population is stable. (hypothetically)
The age of the earth is roughly 4.5 Billion years. Much of that time, it was not colonised, but just to give an upper bound let's assume it was colonized the whole time.
There are 525600 minutes in a year so this give
4.5 ^ 10^9 * 5.3* 10^5 = 2.385 * 10^15 minutes
Dividing by twenty minutes gives 1.1925 * 10^14 total reproductions.
Hence, the number of organisms that ever lived if the entire volume of the crust of the earth were entirely filled with e-coli bacteria reproducing at the maximum of every 20 minutes would be:
(1.2*10^14) * 1.3*10^36 bacteria = 1.56 * 10^50 bacteria
This is an highly exaggerated gross estimate of the maximum number of organisms that ever lived in the extreme best case scenario. In practice it is probably 5-15 orders of magnitude less than this but we get an idea of the kind of numbers we are dealing with.
UPDATE: A more realistic number
The number of bacteria currently on earth is about 5*10^30 forming a biomass which exceeds that of all plants and animals. source.
According to this source cited by the previous wikipedia article:
Hence we have about 1.7 x 10^30 New Cells/year
Multiply this by the estimated 3.5 Billion years (3.5*10^9) when the earth was colonized and we get
For some perspective, the estimated number of atoms comprising all of planet earth is about 1.33 * 10^50 according to this source.
ESTIMATE OF NUMBER OF MAMMALS THAT EVER EXISTED
The upper bound for mammals is as follows.
The surface area of the land of the earth is 149 million square kilometers.
Let us consider if the entire land area of the earth was completely covered with mice.
A mouse is typically 10cm from nose to base of tail and about 5 cm wide=0.1*.05=0.005 square meters. So we get 200 mice per square meter.
Hence, if the entire land area of the earth were covered with mice the world population of mice would be:
149*10^12 square meters * 200 mice per square meter = 1.788 * 10^15 mice
Mice typically produce a litter of 6-8 young and a female can have 5-10 litters per year. source
Let's say we have this maximum population reproducing in a stable manner. i.e. for every new mouse, an old one dies.
Hence the total number of new mice per year would be:
1.8 * 10^15 * 8 young * 10 litters = 1.44*10^17 new mice per year.
Mammals appeared around 200 million years ago (source), so that makes
1.44*10^17 new mice per year * 200 million years
= 2.88 * 10^25 total mice.
In practice mice need space and resources and compete with other mammals which take up space and which reproduce less frequently. Likewise, deserts and polar regions are less inhabitable, and sometimes there are periods of droughts, and they don't always reproduce at the maximum rate etc. so the total number is probably about around 4-10 orders of magnitude less than this.
So it is probably something like 10^20 total mammals ever existed.