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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?

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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.

About 1.9 million species have been discovered and given scientific names, though the actual number may exceed 10 million. Bacteria and archeans could comprise tens of millions species—once taxonomic units are precisely defined. Knowledge about species and extinction rates remains very poor, and species disappear before we know they existed. We propose that, as scientists are better able to assess the conservation status of the species that compose an ecosystem, the more they will understand the health of that ecosystem. It is time to accelerate taxonomy and scientific natural history, two of the most vital but neglected disciplines of biology.

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.

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The question asked for the number of individuals, which is different than the number of species. For reproduction by binary fission, would one (and which one--e.g., the longest survivor?) count as the original individual or are both new individuals? When does a colony or symbiotic collective become a single organism (e.g., mitochondria are presumably not considered distinct individuals but considering the biosphere a single organism seems to be going too far)? The high and highly variable rate of bacterial reproduction would make estimating this count problematic. –  Paul A. Clayton Jan 28 '13 at 15:02
I tried to amend this... but you're right Paul I did misread this. I guess my eyes misled me because the question seems even more impossible. Estimates of bacteria living deep into the earth's crust have seriously multiplied in the past decade or two since they have been discovered and is still being discovered..newscientist.com/article/… Although the question of how many living animals would be more likely, they are hard since only select environments can be surveyed via fossil records. still many dinosaurs, etc to discover. –  shigeta Jan 28 '13 at 18:49
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